SISSA The Higgs as a Supersymmetric Nambu-Goldstone Boson ... Parolini.pdf · Foreword This PhD thesis contains part of the work done by the author in the period 2012-2014 which led

SISSA

Scuola Internazionale Superiore StudiAvanzati

Theoretical Particle Physics Sector

The Higgs as a SupersymmetricNambu-Goldstone Boson

Dissertation in Candidacy for the Degree of Doctor ofPhilosophy

Academic Year 2013-201417 September 2014

Candidate

Alberto Parolini

Supervisor

Prof. Marco Serone

Referees

Prof. Csaba Csaki

Prof. Mariano Quiros

Contents

Abstract 3

Acknowledgements 4

Foreword 5

1 Introduction 6

2 Composite Higgs Models 142.1 The CCWZ Construction . . . . . . . . . . . . . . . . . . . . . 142.2 Composite Higgs Models . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Spurionic Formalism for the Mixing Lagrangian . . . . 202.3 Extra Dimensional Models . . . . . . . . . . . . . . . . . . . . 212.4 The Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Non-SUSY Higgs Mass Estimates . . . . . . . . . . . . 292.5 Experimental probes of Composite Higgs Models . . . . . . . . 32

2.5.1 Direct Searches . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Indirect Measurements . . . . . . . . . . . . . . . . . . 33

3 Supersymmetric Composite Higgs Models 403.1 General Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 General Features of the Higgs potential . . . . . . . . . . . . . 423.3 Minimal Model without Vector Resonances . . . . . . . . . . . 46

3.3.1 Higgs Mass and Fine Tuning Estimate . . . . . . . . . 503.4 Road to Vector Resonances . . . . . . . . . . . . . . . . . . . 533.5 Seiberg Duality for Orthogonal Groups . . . . . . . . . . . . . 54

1

CONTENTS 2

4 A UV Complete Model 594.1 The Basic Construction . . . . . . . . . . . . . . . . . . . . . . 594.2 A Semi-Composite tR . . . . . . . . . . . . . . . . . . . . . . . 634.3 Landau Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4 Lifetime of the Metastable Vacuum . . . . . . . . . . . . . . . 694.5 RG Flow of Soft Terms . . . . . . . . . . . . . . . . . . . . . . 724.6 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.7 Quark Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.8 Flavor Analysis of Dimension Five Operators . . . . . . . . . . 804.9 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 844.10 Detection Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 A Fully Composite Top Right Case 905.1 A Fully Composite tR . . . . . . . . . . . . . . . . . . . . . . . 905.2 Vacuum Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Landau Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.4 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Conclusions and Outlook 97

A Renormalization of the Higgs Potential 99A.1 Renormalization of the Higgs Potential . . . . . . . . . . . . . 99A.2 SO(5) preserving anomalous dimensions and β functions . . . 102

B Unitarization of WW Scattering 105B.1 Minimal Model SO(5)→ SO(4) . . . . . . . . . . . . . . . . . 106B.2 Model with Vector Resonances . . . . . . . . . . . . . . . . . . 107

C SO(N) 110

Bibliography 111

Abstract

This thesis deals with the idea that the Higgs scalar doublet of the Stan-dard Model is a Goldstone boson arising from a spontaneous breaking ofan approximate global symmetry, driven by new physics effects, otherwiseunobserved. This would regulate the behavior of the Higgs potential withthe aim of addressing the Standard Model hierarchy problem, limiting thevalidity of the Standard Model as an effective theory to processes at ener-gies below a cutoff around the TeV scale. After introducing this paradigm,also reviewing some literature, we describe few explicit models, minimal insome sense, built in the context of supersymmetric field theories. We showthe details of the constructions, in particular its novel properties given bythe presence of supersymmetry. We take into account the most importantexperimental constraints and we argue that some versions of these theoriescan be soon tested at collider experiments.

3

Acknowledgements

I am indebted to my supervisor, prof. Marco Serone, for his patient guiding,for his continous incitement, for the possibility to work with him and takingadvantage of his experience and insight of current research in physics. Theworks collected here are the efforts of a pleasant collaboration.

It is also a pleasure to thank the TPP sector of SISSA and other peoplein the Trieste area for several discussions and suggestions, in particular prof.Matteo Bertolini, prof. Andrea Romanino and prof. Giovanni Villadoro.

I am grateful to my colleagues, they contributed to achieve an enjoyableenvironment during these years spent as a PhD student. I wish to expresshere my gratitude for the infinitely many discussions about physics and abouteverything else; among them I would like to mention Francesco Caracciolo,Emanuele Castorina, Bethan Cropp, Lorenzo Di Pietro, Piermarco Fonda,David Marzocca, Marco Matassa, Carlo Pagani, Flavio Porri and MarkoSimonovic.

4

Foreword

This PhD thesis contains part of the work done by the author in the period2012-2014 which led to the articles:

• [1] F. Caracciolo, A. Parolini and M. Serone, “UV Completions of Com-posite Higgs Models with Partial Compositeness”, JHEP 1302 (2013)066 [arXiv:1211.7290 [hep-ph]].

• [2] D. Marzocca, A. Parolini and M. Serone, “Supersymmetry witha pNGB Higgs and Partial Compositeness”, JHEP 1403 (2014) 099[arXiv:1312.5664 [hep-ph]].

• [3] A. Parolini, “Phenomenological aspects of supersymmetric compos-ite Higgs models”, [arXiv:1405.4875 [hep-ph]].

5

http://arxiv.org/abs/1211.7290



Chapter 1

Introduction

The Standard Model (SM) of particle physics is a renormalizable quantumfield theory established in the last fifty years aiming to describe interactionsamong observed particles. Despite its accuracy in describing many observedprocesses there are several experimental hints suggesting that it has to beextended to take into account for other phenomena. The recent observationof a neutral scalar boson with a mass of 126 GeV [4, 5]1 resembling theHiggs boson carried renewed vigor to the, otherwise unverified, hypothesisof the breaking of the electroweak (EW) symmetry through the simplestmechanism, an elementary scalar field charged under SU(2)L×U(1)Y with anon vanishing vacuum expectation value (VEV).

Despite its simplicity the idea of a light elementary field is not appealingfrom a theoretical point of view because light scalars are unnatural. TheSM is no exception: it classically possesses an approximate scale invariancesymmetry broken by the mass term of the only scalar field present, the Higgsfield. This mass term in turn sets the EW scale, the VEV v of its neutral

component, which is experimentally constrained to be v = (√

2GF )−1/2 ' 246

GeV. Quantum fluctuations make the mass square (quadratically) sensitiveto any cutoff of the theory and so a natural theory, in which this quantityis what is expected to be, either has a cutoff at the EW scale or is validon its own at arbitrary high energies. There are a lot of experimental hints

1In the past few months the experimental collaborations provided a measurement withsmaller uncertainties, of the order of the percent; the value of the measured mass is closerto 125 GeV rather than 126 GeV [6,7]. Nevertheless we will continue to refer to 126 GeVbecause most of the work collected here has been carried on before this measurement: inany case the change is small and no conclusion is affected by this new result.

6

CHAPTER 1. INTRODUCTION 7

suggesting that the second cannot be true,2 the strongest among them isnot a hint but a fact: gravity exists and becomes non negligible at energiescomparable with the Planck scale MPl ' 1018 GeV. Therefore in this caseenormous cancellations among unrelated quantum contributions to the Higgsmass occur leading to the observed value(

v

MPl

)2

' 10−32 � 1 , (1.1)

and at the present time we do not have any reason why this should happen.We stress that independently of the new physics entering at MPl the observedvalue of the Fermi scale is extremely unnatural because under the SM RGevolution this solution is highly unstable: if naturalness is restored it has tobe done at a low scale. Otherwise we are led to accept the unnaturalnessof the world in which we live and explain the needed Fine Tuning (FT) asa consequence of some environmental selection: this calls for extensions ofthe SM, as certain constructions from string theory, with a (large) numberof vacua. In each of this vacuum (a universe) v assumes a different valueand we observe only a value of v which is compatible with the existence ofobservers: determine which values are allowed, and which SM parameters (ei-ther dimensionless or dimensionful) are allowed to vary from one universe toanother is a hard task. Without strict assumptions about the detailed struc-ture of the landscape of vacua and assuming that the values for the varyingparameters are distributed with a reasonable probability an efficient mech-anism to fix them is to find conditions related to the presence of observerswhich put bounds (or corners in the case of a multidimensional parameterspace) on these values. For instance consider the case of a single parameterx occurring with a probability density p(x) on the vacua: then

dP

dx= p(x) ⇒ dP

dy= p(ey)ey where y = log x (1.2)

and a flat distribution p′(x) = 0 implies that larger values of x are exponen-tially favored. If at the same time there exists a limiting value xc such that

2An example is the measured value of the physical Higgs mass: 126 GeV fixes thequartic coupling at the weak scale such that it crosses zero at a scale 1011 GeV becomingnegative. Strictly speaking this is not a problem because this running is subject to largeuncertainties and this scale can be shifted to MPl within 2σ. Moreover for the centralvalue the lifetime of the EW breaking vacuum is larger than the age of the Universe [8,9].


universes with x > xc do not lead to the formation and survival of observerswe conclude that most likely we live in a universe with a measured value ofx of the same order of magnitude xc, saturating the upper bound. From thisdiscussion it is already clear that this line of reasoning can be extended to“explain” the value of other SM parameters different from the Higgs VEVv: in fact it was originally proposed for the cosmological constant [10], withseveral refinements as [11, 12], and we can also imagine to generalize it. Ingeneral it is not simple to decide which parameters (and how and why) areanthropically chosen. It is worthwhile mentioning that within the contextof multiverse theories anthropic selection is not the only option and there isthe possibility that some dynamics is responsible for selecting specific val-ues for parameters, see e.g. the discussion of near-criticality in [9]. Finallywe remark that detecting, and therefore confirming, the existence of otheruniverses seems very infeasible. We will not further discuss this possibility.

The other possibility3 is a low energy cutoff close to the EW scale butnot necessarily coinciding with it: taking into account a possible loop sup-pression, the SM as an effective theory should be valid up to the TeV scaleand the resulting theory would be perfectly natural. This second attitudeis not on firm grounds, we can collect a number of arguments pointing tothe direction of a high energy cutoff: flavor measurements, precision tests, apossible gauge coupling unification, neutrino masses through a see-saw mech-anism, inflation. Any physics Beyond the SM (BSM) should take them intoaccount. The most urgent threat to natural physics at the TeV comes fromexperimental searches, mostly at the LHC. Many models studied in the liter-ature predict the existence of new states, typically charged under the SU(3)cSM color gauge group: their presence is related to the dominant role of thetop quark in the loops correcting the Higgs mass in the SM. They should becopiously produced through colored interactions and then decay leaving de-tectable signatures. Currently bounds for stop and gluino masses are around

3To be fair we mention a third way to resolve the dichotomy: if for some reason newphysics, including gravity, does not introduce any mass shift proportional to a thresholdscale a small value of the Higgs mass is natural in the sense that it is stable under RGevolution: this is because its beta function is proportional to the mass itself. Despitethe fact that it is not known if this possibility can be realized it has lately receivedattention, under the names of finite naturalness, UV naturalness, natural tuning andsimilar names. Also, in the context of the minimal supersymmetric SM (MSSM) specificboundary conditions at some high scale, as MGUT , allow for the existence of a focus pointand a Higgs soft mass not scaling with the other soft terms is obtained without a largeFT.


600 GeV and above 1 TeV respectively, while for Kaluza Klein (KK) gluonexcitations the bound is around 2 TeV and finally 800 GeV for composite toppartners with exotic 5/3 electric charge [13]: see the supersymmetry (SUSY)and Exotics webpage of ATLAS and SUSY and Exotica of CMS collabora-tion. These bounds are derived under simplifying assumptions and are notin general valid for any model of BSM physics: on the contrary there aremany explicit models evading these bounds and preserving naturalness andthey are an active research direction. Anyhow in this regard we well resumethe general feeling reinterpreting a quote of E. Fermi: where is everybody?

SUSY surely offers one of the best motivated possibilities to solve thehierarchy problem in a natural way. Relating the number of bosonic andfermionic degrees of freedom in the UV allows for cancellations in the loopcontributions. The Minimal Supersymmetric Standard Model is the simplestSUSY generalization of the SM: it introduces SUSY partners for every SMfield and it contains one Higgs doublet more, necessary for anomaly cancella-tions. Furthermore it has many other welcome features, namely it improvesgauge coupling unification and it provides a dark matter candidate. Never-theless the Higgs boson mass is predicted to be too low: at tree-level it hasto satisfy the bound

M2H ≤ m2

Z cos2 2β (1.3)

where tan β measures the relative importance of the two Higgs doublet inparticipating to the EWSB mechanism, it is the ratio of the two VEVs.Loop corrections are important and necessarily depend on SUSY breakingparameters, they can increase the Higgs mass at the price of a larger amountof FT, which is around 1% or less in the MSSM [14]. At tree-level the largestsource of FT is given by higgsino masses, governed by the so called µ term ofthe superpotential; at one-loop also stop soft parameters enter and finally, attwo-loops, gluino masses are responsible, if too heavy, for a certain amountof FT. The situation gets improved for non minimal extensions, new (su-per)fields with new interactions can increase the Higgs mass. For instancethe Next to Minimal Supersymmetric SM (NMSSM) is just the MSSM cou-pled to a chiral superfield, gauge singlet, with appropriate superpotentialinteractions.

Moreover there is a more profound reason why the MSSM is not enough,and it is linked to the SUSY breaking. The breaking has to be soft, namelyonly relevant operators are allowed, such that the UV nice behavior of thetheory is not spoiled. But we need other fields spontaneously breaking SUSY

https://twiki.cern.ch/twiki/bin/view/AtlasPublic/SupersymmetryPublicResults

https://twiki.cern.ch/twiki/bin/view/AtlasPublic/ExoticsPublicResults

https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsSUS

https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsEXO


and we have to make sure that there exist interactions4 among this new sectorand the MSSM fields, the visible sector, such that SUSY breaking is inducedalso for visible fields, namely SUSY particles are much heavier than theirordinary partners, satisfying at the same time other constraints for instancefrom flavor physics. The necessity of a separate sector responsible for SUSYbreaking comes from the supertrace mass formula, which relates the sum ofthe squared masses of all the particles in the spectrum, properly weightedaccording to their spin, to the D terms computed in the vacuum: it cannotbe satisfied with only the MSSM fields.

Finally we stress that SUSY can be broken either at tree-level or bynon perturbative effects. This second option is favored because it dynami-cally generates a hierarchy among mass scales. A very generic expectationis a SUSY theory spontaneously broken, by non perturbative effects, in ametastable vacuum. A central role is played by the R symmetry, a bosonicU(1) whose generator does not commute with the supercharges. It forbids(Majorana) gaugino masses and therefore it has to be broken but, on theone hand, if it is spontaneously broken in the vacuum it implies the presenceof a too light R axion, the associated Goldstone boson; on the other handit can be shown that in a generic theory without an R symmetry SUSY isnot broken [15]. However it can be that we start with a theory without anR symmetry and besides one or more SUSY preserving vacua there existsa metastable vacuum, a local minimum of the potential, breaking SUSY. Inthis vacuum there exists an approximate R symmetry which still allows forgaugino masses, but the full theory does not respect it. If its lifetime is longenough it provides a viable way out of the aforementioned tension.

Composite Higgs Models (CHM) are another possible extension of theSM of the second attitude mentioned before, aiming to solve the naturalnessproblem in a natural way, namely introducing new physics at the TeV scale.The general idea is that the Higgs is not an elementary scalar but insteada bound state of some new objects charged under a new, still unknown,strongly coupled gauge theory. This is similar to technicolor theories in whicha condensate breaks the EW symmetry SU(2)L × U(1)Y to the U(1)Q: thesimilarity is in having a new gauge interaction responsible for this breaking;the main difference is that in CHM there exists a light scalar in the spectrum,with the right quantum numbers, whose VEV breaks the EW group. Several

4Gravity will eventually couple the two sectors. One might want, for several reasons,to avoid relying on it and having some other mechanisms of mediation.


different realizations of this idea have been studied and they can be classified:an up to date review, containing also several other interesting discussions, isgiven in [16]. In this thesis we concentrate on models of Higgs as a pseudoNambu Goldstone bosons (pNGB): we do not discuss little Higgs or otherrealizations. A doubtless elegance of these models is in the dynamical originof EWSB, whereas many others models, as for instance the MSSM, simplyaccept it as a matter of fact item.

As mentioned before any light scalar is unnatural because its mass isadditively renormalized. Moreover scalar bosons are not protected by anysymmetry and the massless limit does not restore any symmetry. This is notthe case for Goldstone bosons of a spontaneously broken global symmetry:in this case the broken symmetry is non linearly realized and the associatedGoldstones transform under shift. Therefore they do not have non derivativeinteractions and they are massless: in fact they are so constrained that theirlow energy Lagrangian, a non linear σ-model, is determined only by thestructure of the coset associated to the breaking, regardless any other detail[17,18]. Identifying the components of the Higgs doublet as Goldstone fieldswould therefore predict their mass to be exactly zero. Since this is notthe case we want the global symmetry spontaneously broken to be only anapproximate symmetry of the theory, namely we introduce some terms ofexplicit breaking: these terms drive a non trivial profile for the would beGoldstones. This radiatively induced potential is responsible for the EWbreaking providing a non vanishing VEV for the Goldsones identified withthe Higgs.

The typical situation is as follows: a gauge theory is responsible, throughnon perturbative effects, to the breaking of a global symmetry G to a propersubgroup H: the scale associated, denoted by Λ, is dynamically generatedand exponentially suppressed with respect to the scale where the theory isdefined perturbatively, for instance MPl. The scale f < Λ, the decay con-stant of the σ-model, regulates the interactions of the associated NambuGoldstone bosons. The unbroken symmetry H has to contain the SM groupSU(3)c×SU(2)L×U(1)Y : the broken generators have to be charged under theEW group SU(2)L ×U(1)Y in order to be identified with the Higgs doublet.This means that the symmetry G is explicitly broken by a partial gauging:moreover the coupling to SM fermions, in particular the top, is also explic-itly G violating. This explicit breaking is mediated by a mechanism namedPartial Compositeness (PC): SM fields acquire small components in termsof bound states of the new sector, in contrast to the Higgs which is fully


composite. The resulting neat effect is a potential for the Goldsone bosons:it provides a VEV v < f to the neutral component of the Higgs. The ratio

ξ =v2

f 2(1.4)

quantifies the separation of scales occurring among the Higgs and new physics.While ξ is generically expected to be O(1) a certain separation is requestedfrom phenomenological reasons: therefore ξ encodes a measure of the FT ofthe model. In the following we focus on the benchmark value

ξ = 0.1 . (1.5)

In the models which we are going to discuss the FT is estimated in arefined way and it is computed numerically using the definition of [19] withlogarhitmic derivatives: the actual level of FT is found around 1− 2%.

Given the negative results so far obtained for any BSM physics search,and insisting on naturalness, it is more than worthwhile to explore less mini-mal scenarios and to pursue new ideas. Therefore it has been natural to studysupersymmetric realizations of pNGB CHM; the benefits from the union ofthe two frameworks well studied in separation, SUSY and composite Higgs,are manifold as we will explain in detail: SUSY provides tools to deal withstrongly coupled theories, namely it allows us to build explicit models whoserange of validity as effective field theories is not limited by non perturba-tively generated energy scales5, and it helps in controlling the little hierarchyproblem generically affecting BSM theories; at the same time stops or othersuperpartners are not already excluded and the same is true for fermionicresonances, while they are within the reach of Run II at LHC in their lesstuned versions.

Pure four dimensional UV completions of the CHM paradigm have beenrecently investigated also in [20–22], focusing on constructions without ele-mentary scalars. On the other hand holographic descriptions, based on fivedimensional spacetime, of the strongly coupled gauge theory have been ex-ploited: we will comment on them in the next chapter. SUSY allows, namelythanks to Seiberg duality, a greater level of calculability, and therefore predic-tivity, than generic gauge theories, without enlarging the number of spatialdimensions.

5We often encounter Landau poles at high energies, therefore a UV cutoff must existat some scale below MPl; the interesting part is that its appearance is not strictly relatedto the strong dynamics producing the pNGB Higgs.


The rest of the thesis is organized as follows: in chapter 2 we introduceCHM based on a pNGB Higgs, we discuss partial compositeness, the Higgspotential, the main features and some experimental constraints; from chap-ter 3 on we concentrate on original results and we turn to supersymmetricmodels, we again analyze the Higgs potential on general grounds and wesequently focus on one specific model in some sense minimal. The param-eter space is inspected with the help of numerical scans. In chapter 4 wemove to another model with a richer gauge structure, reviewing the mostimportant bounds from negative direct detection of heavy states combinedwith a numerical scan. We review how to deal with the RG flow of UV softSUSY breaking terms in presence of confining dynamics. Since the modelsits in a metastable vacuum we address the issue of its decay and we showthat a lifetime larger than the age of the universe is fully compatible with allthe other requirments. We also discuss a mechanism, different than partialcompositeness, to transmit EWSB to all SM fermions (we do it explicitly forquarks): we rely on dimension five operators in the initial superpotential. Wecheck also that they satisfy bounds coming from flavor observables, mainlyfrom mesons physics. Chapter 5 is devoted to the study, both analytical andnumerical, of an excluded model in which the right top is fully composite: thediscussion proceeds, with less details, in resemblance to the one carried onin the prior chapter. Finally we draw our conclusions and we close the thesiswith few appendices. In appendix A we collect some one-loop beta functionscoefficients on which we rely in the main text and we explicitly show howthey enter the scale dependence of the Higgs potential. In appendix B westudy how the non linearities of the Higgs field and new massive resonancesmodify the high energy behavior of the two to two scattering of EW bosonsin the two models discussed in the text: in particular we show that pertur-bative unitarity of the amplitude of the process is recovered. We report ourgroup matrices conventions in appendix C.

Chapter 2

Composite Higgs Models

This chapter is devoted to introducing the pNGB Higgs. We begin review-ing the Callan Coleman Wess Zumino (CCWZ) description of low energyLagrangian for Goldstone bosons in a generic theory and we employ it to-gether with some general parametrization of the strong sector breaking thesymmetry. We describe the interactions with the SM and how they generatea potential for the Higgs, outlining its most relevant features. We also makecontact with extra dimensional theories, also via the AdS/CFT correspon-dence. Finally we survey the experimental constraints on these models, bothfrom direct searches and indirect measurements.

2.1 The CCWZ Construction

We briefly recall the CCWZ construction of the low energy Lagrangian forGoldstone bosons [17,18]. Given a global symmetry (Lie) group G broken toa subgroup H we distinguish the generators of the associated algebra amongunbroken T i and broken Xa. They satisfy the following

[T i, T j] = if ijkT k, [T i, Xa] = if iabXb, [Xa, Xb] = ifabcXc + ifabiT i .(2.1)

The structure constants f are antisymmetric in their indices. We work withorthogonal and unitary groups and a simple realizations for the generators isin terms of hermitian matrices. If the constants fabc vanish the coset is saidto be symmetric and its Lie algebra is invariant under a discrete Z2 symmetryunder which T i are even and Xa are odd; we call R this automorphism of

14

CHAPTER 2. COMPOSITE HIGGS MODELS 15

the algebra of G:R(T i) = T i, R(Xa) = −Xa . (2.2)

In a neighborhood of the identity any g ∈ G can be decomposed as

g = eiξ·Xeiu·T (2.3)

such that∀ g0 ∈ G g0e

iξ·X = eiξ′·Xeiu

′·T (2.4)

where ξ′ and u′ are appropriate functions of ξ and g0. With the definitionU = eiξ·X this last can be expressed as

U → g0Uh−1(ξ, g0) . (2.5)

We focus on the product

U−1∂µU = iXaDaµ + iT iEi

µ (2.6)

where the decomposition along the generators is meaningful because the l.h.s.is an element of the algebra of G. Under a transformation g0 ∈ G

U−1∂µU → h(U−1∂µU)h−1 − (∂µh)h−1 . (2.7)

From this we can read:{Dµ = XaDa

µ → hDµh−1

Eµ = T iEiµ → h(Eµ − i∂µ)h−1 . (2.8)

With these objects we can build a Lagrangian invariant under a linear andlocal H symmetry, where Eµ acts as a gauge field and we can build out of itcovariant derivatives and a field strength. This Lagrangian is also symmetricunder any transformation g0 ∈ G, which acts non linearly. In an expansionin the number of derivatives the leading term is given by

1

2f 2Da

µDaµ , where Da

µ = ∂µξa + . . . (2.9)

One can prove that under a transformation generated by broken Xa, ξ trans-forms with a shift; moreover ξa are identified with the canonically normalizedGoldstone bosons, ξa = f−1πa.


For symmetric coset space the matrix U2 turns out to be useful becauseit transforms linearly under G: it can be proven that

U2 → gU2R(g)−1 (2.10)

where R is the automorphism defined in eq.(2.2).With this procedure of constructing invariant Lagrangians one can miss

terms which are not invariant but shift by a total derivative under a symmetrytransformation [23,24]. These terms in the action can be written as

SWZW =

∫S4

β (2.11)

where S4 is the (compactified) spacetime. If β shifts by a total derivative un-der a transformation symmetry the 5-form ω = dβ is left invariant. Becauseof the Stokes theorem we can also write

SWZW =

∫S4

β =

∫Bω (2.12)

such that ∂B = S4. Therefore the allowed invariants are related toH5(G/H,R),the fifth cohomology class of the coset G/H [25]. However for the only cosetwe are going to discuss, SO(5)/SO(4), this object vanishes and none of theseterms can be added to the effective Lagrangian.

Finally the present discussion can be generalized to the case of a gaugedsubgroup of G: the simplest way is to gauge the entire group G and discardthe non dynamical fields at the end, consistently replacing derivatives withcovariant ones and adding kinetic terms for the field strengths. As expectedthe Goldstone bosons along the broken gauged directions are eaten.

One may be interested in including in the low energy Lagrangian alsoother fields, massless or massive. Their interactions are not completely fixedby symmetry considerations, nevertheless they can be included in a G invari-ant way if they transform in a proper way under the unbroken subgroup H,namely if they fullfill representations of H: it is sufficient to note that withthe help of the matrix U we can “lift” them to representations of G; thisfact is simple to understand recalling eq.(2.5). From a top down perspective,knowing a UV Lagrangian spontaneously breaking G → H, the redefinitionof all the fields through the matrix U makes the Goldstones disappear fromthe non derivative part of the Lagrangian. In our context this operation isuseful to treat the explicitly G breaking terms.


An outstanding example of the use of the CCWZ formalism is the con-struction of the pion Lagrangian in QCD (with two quarks only, up anddown): it presents analogies with the case of composite Higgs. Pions arethe Goldstone bosons associated to the spontaneous breaking of the chiralsymmetry SU(2)L×SU(2)R → SU(2)V triggered by non perturbative effects.This formalism allows to overtake the difficulties of the strongly interactingQCD and we correctly identify the low energy degrees of freedom, the pions.The Wess Zumino Witten term, described in eq.(2.11) and eq.(2.12), for theQCD is responsible for a term in the Lagrangian of the form

L ⊇ − Nce2

48π2fππ0εµνρσFµνFρσ (2.13)

where Nc = 3 is the number of QCD colors and fπ is the pion decay constant.This describes an interaction vertex among the neutral pion and photons andit is related to the decay process π0 → 2γ. Moreover the chiral symmetry isnot a true global symmetry of the QCD Lagrangian: it is explicitly broken byquark masses and by the gauging of the electromagnetic group U(1)Q. Thesebreaking generate a potential for the pions, otherwise forbidden. Indeedpions in nature are not massless particles. Quark masses

L ⊇ muuu+mddd = qMq (2.14)

can be written in a way formally respecting the chiral symmetry, namelypretending that the matrix M transforms as a bidoublet, M → LMR† actingwith an element of SU(2)L×SU(2)R. Imposing formal chiral invariance, withthe help of this spurious mass matrix, we can argue that the low energyLagrangian will contain

L ⊇ cf 3πTr[MU ] (2.15)

where U contains the three pions π± and π0, U = exp(ifσaπa

), a = 1, 2, 3

and Tr[σaσb] = δab. Expanding for large fπ we obtain a mass term for thepions,

L ⊇ − c4fπ(mu +md)π

aπa . (2.16)

A more detailed computation leads to the exact determination of the pionmasses, c = 〈qq〉

f3π. The presence of the U(1)Q further breaks the chiral sym-

metry and the electric charge is responsible for the mass difference amongthe charged pions and the neutral one.


2.2 Composite Higgs Models

The main ingredient of pNGB CHM is a strongly interacting sector which weconsider at first in isolation from the SM: it contains the new physics showingup at the TeV scale. We frequently refer to it as composite sector, in contrastto an elementary sector which contains the SM fields, to stress that in thecomposite sector a new gauge interaction is at work, resulting in bound states;more in general, with the aim of a unified discussion, we will continue todenote as composite the sector responsible for the pNGB Higgs also in caseswhere the fields are not composite objects. Original models were formulatedin terms of a conformal field theory (CFT) and the relevance of holographictechniques have been extensively stressed [26]. They are characterized by twomass scales: starting from a UV point, either free or interacting, the theoryflows to an IR scale, hierarchically suppressed, where it becomes stronglycoupled and an operator breaks a global symmetryG to a proper subgroupH.Before entering the details of the construction we note en passant the analogywith the chiral Lagrangian for pions described at the end of section 2.1: inthat case the symmetry breaking pattern is SU(2)L×SU(2)R → SU(2)V andwe have three Goldstone bosons, the pions; we instead focus on cosets withfour Goldstone and we identify them with the Higgs. In the case of QCD wehave mass terms for quarks and the electromagnetic interactions explicitlybreaking the chiral symmetry, while in our pNGB Higgs we introduce massmixings, under the name of partial compositeness, and we gauge the EWgroup, again bringing an explicitly breaking: in turn it induces a potentialfor the Goldstones. The main difference is that in QCD the pion masses arepositive, here we obtain a negative (squared) mass for the Higgs, since it hasto trigger EWSB.

In the literature several models have been extensively explored, for in-stance SU(N)/SO(N) and SO(N)/SO(N−1). We focus on the simplest real-ization allowing for a custodial symmetry with G = SU(3)c × SO(5)×U(1)Xand H = SU(3)c × SO(4) × U(1)X : the interesting part of the breaking isin SO(5) → SO(4) and the other factors are spectators, nevertheless theyare needed for a realistic model1. The four Goldstone bosons live on thecoset SO(5)/SO(4) ≡ S4 and they are identified with the Higgs doubletof the SM: therefore we realize the SM symmetries gauging a subgroup of

1Strictly speaking G and H can be larger, the key point is that they contain SO(5)×U(1)X and SO(4)× U(1)X respectively.


the global symmetry of the composite sector: we recall the isomorphismSO(4) ' SU(2)L×SU(2)R and we identify the hypercharge as T 3

R+X, wherethe X can be seen as proportional to the SM B −L; the choice for the colorgroup is self evident. With this identification the real fourplet in the 4 ofSO(4) has the right quantum numbers, it is a composite state of the newphysics sector and it has a vanishing potential: it is a good candidate toplay the role of the Higgs doublet. Moving to the realistic case with the SMcoupled the total Lagrangian schematically is

L = Lcomp + Lel + Lmix . (2.17)

The first term in the Lagrangian is coming from the strongly interactingtheory and it contains, among others, the kinetic term for the pNGB Higgsfrom eq.(2.9). The second term, elementary in contrast to the composite one,is the Lagrangian for the SM fields without the Higgs, namely the kineticterms for fermions and gauge bosons. These fields are neutral under the newstrong gauge group and therefore they do not participate in the formation ofbound states. The third part is the mixing part and it has the form

Lmix = gaJaµA

aµ + · · ·+∑r

εrψrOr + h.c. . (2.18)

The first part contains the currents of the composite sector associated tosymmetries gauged by the SM, where the dots include possible terms withhigher powers of the gauge fields; the second part is a sum of portal interac-tions for SM fermions, in principle ψr ∈ {qL, uR, dR, lL, eR}, with compositeoperators Or with suitable quantum numbers. The energy scaling of εr’s aredictated by the anomalous dimensions γr ' dim(Or)− 5

2: the scaling dimen-

sions of Or can be significantly different from the classical ones and thereforeeach of these mixings εr can be driven weak or strong in the IR. Because ofthem the SM fermions are partially composite where the degree of compos-iteness varies for different r, and for different families, and it regulates thestrength of their coupling to the Higgs boson: the mass eigenstates comingfrom larger mixings have a more significant component in terms of compos-ite states and they couple more to the Higgs, therefore they are heavier. Itis clear that the more composite SM fermion is the top and we expect theassociated anomalous dimensions γ’s to be the smallest.

Partial compositeness, introduced in [27], is studied in a simplified versionin [28]. It consists in treating Or as vector-like (Dirac) massive fermions and


declaring that εr are mass mixings: the relevant part of eq.(2.17) can bewritten as

L = ψri/∂ψr + Or(i/∂ −Mr)Or + εrψrOr + h.c. (2.19)

The mass eigenstates are given by(cosφr sinφr− sinφr cosφr

)(ψrPrOr

)(2.20)

where Pr projects onto the suitable chirality state and the mixing angle is

tanφr = εrMr

; the mass eigenvalues are 0 and (M2r + ε2r)

1/2. We understand

this approximation looking at the two point function 〈Or(q)Or(−q)〉: in thelarge N limit of a gauge theory we expect it to be the sum of narrow reso-nances with increasing masses and the above simplification is nothing morethan the truncation to the first resonance; alternatively we may think torotate to a basis where only one state, approximatively the lightest, has amixing with the elementary ψr. An analogous reasoning line applies to vectorcurrents and heavy spin-1 resonances.

In the SUSY models we will discuss the operators responsible for fermionicpartial compositeness are marginal in the UV, where the BSM is weaklycoupled, and they flow to relevant deformations. They are mesonic, ratherthan baryonic, fermion states of the confining gauge theory: their presenceis well understood thanks to SUSY and Seiberg duality.Lmix is explicitly SO(5) breaking and therefore the degeneracy of the

Higgs is lifted. The potential generated by gauge interactions does not triggerEWSB because it tends to align the vacuum in a preserving direction [29],therefore the contribution of the matter fields, mainly the top fermion, iscrucial. We will elaborate more on this compensation of effects as one of thesources of the FT.

2.2.1 Spurionic Formalism for the Mixing Lagrangian

The mixing Lagrangian eq.(2.18) can be made formally SO(5) invariant en-larging the symmetry to SO(5)×U(1)X × SU(2)0,L ×U(1)0,R ×U(1)0,X ; theoperators Or transform in SO(5) × U(1)X representations, the SM fields ψrare charged under the other SU(2)0,L×U(1)0,R×U(1)0,X factors and the cou-plings εr’s are promoted to spurions charged under both symmetries: theirVEV break the symmetry to a diagonal combination.∑

r

ψr · εr ·Or + h.c. . (2.21)


Similarly promoting the SM gauge coupling g and g′ to spurions allows torecover SO(5) invariance [30]. Alternatively, but completely equivalently, wecan embed the elementary fields ψr in spurionic representations of SO(5) ξrwhose extra components are set to zero. In this way again the invariance isformally recovered: ∑

r

εr ξrOr + h.c. . (2.22)

ξr is chosen such that the product with Or contains the singlet, therefore itdepends on the details of the composite sector. Typically the most studiedrepresentations are the smallest ones which are 4, 5, 10 and 14 for SO(5) [31].The neat examples of the following chapters are based exclusively on the 5.

If we redefine all the fields in the composite sector as explained at theend of section 2.1 with the aim of singling out the Goldstones we remove theHiggs dependence from the non derivative part of the Lagrangian and theonly places where it remains are the SO(5) breaking terms of Lmix eq.(2.18).

2.3 Extra Dimensional Models

In the following we will move to study the main predictions of CHM butbefore doing it we comment on extra dimensional theories, in particular onpossible extensions of the SM defined on a five dimensional spacetime whereone of the spatial dimensions is compactified and we explain their relationwith CHM introduced in the previous sections. Extra dimensional modelshave provived many calculable examples in particular as holographic descrip-tions of the strong sector postulated in section 2.2. With the language offeredby holography extra dimensional theories and strongly coupled CFT are twoequivalent ways of describing the same physics.

The simplest example is R1,3 × S1, although it turns out that the choiceof the orbifold R1,3 × S1

Z2is much more convenient and interesting. The first

consequence of orbifolding the fifth dimension is that proper boundary con-ditions allow to introduce a chiral fermion field, despite the fact the thespinorial representation of SO(4, 1) is not reducible, that is a fermion in fivedimensions is defined with both its left and right chirality components.

There is a second advantage from compactifying the extra dimension onan orbifold, again coming from the imposition of adequate boundary con-ditions: in the case of gauge theory we can reduce the gauge group at low


energy. In particular on the branes gauge invariance is associated to Neu-mann boundary conditions for the corresponding vector field and a five di-mensional gauge theory can be reduced in two different ways at the orbifoldfixed points.

The description of fields in extra dimensional theories and their reductionto four dimensions take great advantage from the Anti de Sitter/ConformalField Theory (AdS/CFT) correspondence. This conjectured correspondencehelps in making contact between extra dimensional models and CHM, itprovides a description of a BSM CFT in terms of a gravitational dual veryoften more calculable than the original field theory. Therefore in the followingwe introduce and briefly discuss warped spaces2, and the correspondence andfinally we comment back on CHM models.

In a theory with n extra dimensions the four dimensional Planck scaleMPl is related to the 4 + n one M by

M2Pl = Mn+2Vn (2.23)

where Vn is the volume of the compact extra dimensions and typically Vn ∼ rncfor a compactification radius rc. In [34] it has been considered a setup,universally known as RSI, of a five dimensional theory with the topologyof R1,d−1 × S1

Z2: at the fixed points of the orbifold two 3 branes are placed,

supporting four dimensional field theories. On these branes an SO(3, 1) ⊆SO(4, 2) isometry group survives.

It has been shown that, with a proper choice of the vacuum energies de-fined in the bulk and on the branes, the resulting spacetime has the geometryof a slice of an Anti de Sitter (AdS) space with a curvature function of Mand another mass scale k < M . The compactification radius rc is a freeparameter but sensible constructions require

MPl > M > k >1

rc. (2.24)

The condition k < M ensures the stability of the AdS solution against quan-tum gravitational corrections. The metric can be explicitly written (choosinga mostly minus signature) as

ds2 = e−2krc|y|gµνdxµdxν − r2

cdy2 (2.25)

2Notice that holographic models with a flat extra dimensions have been also studied,see e.g. [32, 33].


where y parametrizes the fifth dimension, y ∈ [−π, π] and the 3 branes arelocated at y = 0 and y = π, the orbifold fixed points; we emphasize theconsequences of the non factorizability of this geometry. Eq.(2.23) becomes

M2Pl =

M3

k[1− e−2krcπ] ' M3

k(2.26)

and therefore M < MPl. The exact value of rc does not affect the fourdimensional Planck scale but it has a remarkable consequence: it enters thetheory living at y = π through the factor e−2krcπ from the restriction of themetric to the brane, after properly rescaling the fields of the bulk action. Amass term v is redshifted to

v′ = e−krcπv . (2.27)

For a generic relevant operator of dimension α we expect the coupling λ tobe sent to λ′ = e−krcπ(4−α)λ. In this way a value krc ∼ 10 naturally generatesa hierarchy of 15 orders of magnitude among a fundamental mass scale andthe physical one: this is the crucial observation employed to resolve the SMhierarchy problem.

The correspondence AdS/CFT has been originally formulated in the con-text of superstring theory but it is believed to have a wider range of appli-cability. The claim is that a gravitational theory on a warped backgroundis dual to a gauge theory living on a manifold with one dimension less, typ-ically the boundary of the manifold of the gravitational theory, see [35] andalso [36, 37]. Given an operator belonging to the CFT there exists a bulkfield ϕ, i.e. living on the entire five dimensional manifold, with a definedboundary condition ϕ0 such that the gravity solution is unique: from bothlanguages we can extract, in principle, the same amount of information sim-ply relating the effective action for this field and the generating functional ofthe correlators of O in the CFT,

e−Γ[ϕ0] = 〈e−∫

d4xϕ0O〉 (2.28)

(in Euclidean spacetime). The boundary field acts as a source for the operatorO and it allows to compute n point functions in the strongly coupled regime,typically a large N limit in case of a gauge theory, as functional derivativesof the on shell action for the field φ0.

For the holographic interpretation of the RSI scenario we follow [38, 39].The AdS metric eq.(2.25) can be expressed as

ds2 =1

(kz)2 (gµνdxµdxν − dz2) (2.29)


with the change of variable kz = ekrcy. The orbifold fixed points are at thepositions z = 1/k and z = ekrcπ/k. The fifth coordinate z is related to theRG scale of the CFT, therefore we refer to the branes as UV and IR branerespectively. The presence of the UV brane acts as a cutoff for the fieldtheory. Moreover at the scale MPl the CFT couples to gravity, in a way suchthat conformal symmetry is broken only by Planck suppressed operators. Asit is clear from eq.(2.26), written as

MPl =

(M

k

)3

(1

z2UV

− 1

z2IR

) . (2.30)

the limit zUV → 0 implies a CFT valid up to arbitrary high energies andcompletely decoupled gravity, because it implies MPl → +∞. An AdS back-ground modified by the presence of a single brane has been considered in [40],whose scenario is known as RSII. In [40] indeed it has been shown that thisextra dimensional setup of AdS5 ending to a brane supports a massless nor-malizable graviton and a continuum of KK states. Remarkably this alsoshows that, with the only purpose of building a theory consistent with New-ton’s law and general relativity, extra dimensions do not need to be compact.

The addition of the second IR brane introduces a new boundary conditionat zIR = ekrcπ/k and quantizes the spectrum of KK excitations. Moreover theIR brane signals a departure from AdS background at low energies, thereforein the field theory language we expect a breaking of conformality in the IR.The position of this brane is the VEV of a dynamical radion field which isviewed as the dilaton, the Goldstone boson of the broken dilatation symme-try. This VEV can be adjusted, namely the position of the brane can be fixedand the radius of the extra dimension is stabilized, with a mechanism pro-posed in [41]: integrating over an additional single scalar field forced to havea nontrivial profile along the fifth dimension provides a potential for the ra-dion. In the spirit of AdS/CFT correspondence this field is dual to a slightlyrelevant operator which deforms the CFT and it grows in the IR where itcauses the breaking of conformal symmetry, and the hierarchy between scalesis generated through dimensional transmutation.

The holographic interpretation of the RSI can be extended also to gaugefields, including the breaking obtained imposing proper boundary condi-tions [42], and to fermions [43]. In particular while in the original RS modelsthe SM fields where localized, namely confined to live, on the IR brane, it hasbeen realized the convenience of having fields propagating in all the dimen-sions [44]. The bulk mass of a fermion is related to the strength of the mixing


of the associated massless four dimensional fermion with an operator of theCFT. In the language of the KK reduction this comes from the localization ofthe zero mode in the fifth dimension controlled by its bulk mass. A fermionwhose profile is localized close to the UV brane is less sensitive to what hap-pens in the bulk, i.e. the mixing with the CFT is weaker, and therefore werefer to it as more elementary, in contrast to a fermion more localized towardthe IR brane, more related to the strong dynamics. The four dimensionalYukawa couplings are determined by the overlap in the fifth dimension ofthe wavefunctions of the fermions (left and right) with the one of the Higgsscalar.

2.4 The Higgs Potential

Regardless on the possible UV completion we try to draw conclusions on theshape of the generated Higgs potential, not relying on the details of the CFT:we make model independent statements, assuming the coset SO(5)/SO(4)and partial compositeness. The one-loop Higgs potential can be obtainedin a two steps procedure, following for instance [45]: all the new physics isintegrated out and can be encoded in form factors for SM fields; then theone-loop action is computed in the background of the Higgs. Once the strongsector has been integrated out we are left with a Lagrangian (in momentumspace) for the SM fields coupled to the Higgs, specifically the top and theEW gauge bosons:

L = tL /pΠtLtL + tR /pΠtRtR − (tLΠtLtRtR + h.c.) + (2.31)

+P µνt

2

(2ΠW+W−W

+µ W

−ν + ΠW 3W 3W 3

µW3ν + ΠBBBµBν + 2ΠW 3BW

3µBν

)where P µν

t = ηµν − pµpν

p2. We neglect SM fermions different from top be-

cause their contribution is negligible, but this formalism can be straightfor-wardly extended to account for them. The Higgs dependence, not manifestin eq.(2.31), is in the various form factors:

Π = Π(h) . (2.32)

The computation of the effective action is best performed in the Landaugauge where the ghosts and the longitudinal components of the vectors de-couple: the resulting potential does not depend on the choice of the gauge.


A standard one-loop computation leads to

V (h) = −2Nc

∫d4q

(2π)4 log[q2ΠtLΠtR + |ΠtLtR |

2]+ (2.33)

+3

2

∫d4q

(2π)4

[2 log ΠW+W− + log

(ΠW 3W 3ΠBB − Π2

W 3B

)]where every form factor Π is a function on −q2, Π(−q2), and q is the Eu-clidean four momentum; Nc = 3 is the number of QCD colors. In order tosimplify our notation considerably, we work in the unitary gauge and denoteby h the Higgs field in this gauge.

Since the Higgs is a pNGB associated to an approximate spontaneoussymmetry breaking, its VEV is effectively an angle. For this reason it isoften convenient to describe its potential not in terms of the Higgs field hitself, but of its sine:

sh ≡ sinh

f, (2.34)

where f is the Higgs decay constant. Following a standard notation we alsodefine

ξ ≡ 〈s2h〉 . (2.35)

The electroweak scale is fixed to be v2 = f 2ξ ' (246 GeV)2. We focus onsmall values of ξ and in explicit results we set it to the benchmark valueξ = 0.1. Due to the contribution of particles whose masses vanish for sh = 0(such as the top, W and Z), the one-loop Higgs potential contains non-analytic terms of the form s4

h log sh that do not admit a Taylor expansionaround sh = 0. In the phenomenological regions of interest, these termsdo not lead to new features and are qualitatively but not quantitativelynegligible. However, they make an analytic study of the potential slightlymore difficult.

For sh � 1, the tree-level + one-loop potential V = V (0) + V (1) admitsan expansion of the form

V (sh) = −γs2h + βs4

h + δs4h log sh +O(s6

h) (2.36)

where

γ = − 1

2

∂2V

∂s2h

∣∣∣∣sh=0

, δ =sh4!

∂5V

∂s5h

∣∣∣∣sh=0

, β =1

4!

∂4(V − δs4h log sh)

∂s4h

∣∣∣∣sh=0

(2.37)


and higher order terms are entirely neglected. In a naive expansion aroundsh = 0 the presence of δ would be detected by the appearance of a spuriousIR divergence in the coefficient β. At first order in δ around δ = 0 the nontrivial minimum of the potential, assuming it exists, is found at

〈s2h〉 ≡ ξ = ξ0

(1− δ

4β(1 + 2 log ξ0)

), (2.38)

whereξ0 =

γ

2β(2.39)

is the leading order minimum for δ = 0. The physical Higgs mass, computedas the second derivative of the potential at its minimum, is given by

M2H =

8β

f 2ξ0(1− ξ0) +

4δξ0

f 2

(1− ξ0

2+ ξ0 log ξ0

). (2.40)

For ξ0 � 1 we get

M2H ' (M0

H)2(

1 +δ

2β

), (2.41)

where

(M0H)2 ' 8β

f 2ξ0 . (2.42)

In the limit δ = 0 we would recover

ξ =γ

2β, M2

H =8β

f 2ξ(1− ξ) . (2.43)

In all the models we will consider, there are two distinct sectors that do notcouple at tree-level at quadratic order: a “matter” sector, including the fieldsthat mix with the top quark and a “gauge” sector, including the SM gaugefields and other fields, neutral under color. The matter and gauge sectorscontribute separately to the one-loop Higgs potential:

V (sh) = Vgauge(sh) + Vmatter(sh) (2.44)

and

γ = γgauge + γmatter , β = βgauge + βmatter , δ = δgauge + δmatter . (2.45)

The explicit SO(5) symmetry breaking parameters are the SM gauge cou-plings g and g′ in the gauge sector and the mixing parameters εr’s given by


eq.(2.18) in the matter sector. Since the latter are sizably larger than theformer, for sensible values of the parameters βmatter � βgauge.

3 At fixed ξ,then, the Higgs mass is essentially determined by the matter contribution (inthe numerical study, however, we keep all the contributions to the one-looppotential). The gauge contribution should instead be retained in γ becausethe FT cancellations needed to get ξ � 1 might involve γgauge.

In the models we considered, the particles massless at sh = 0 are alwaysthe top in the matter sector and the W and the Z gauge bosons in the gaugesector. Correspondingly, the explicit form of δ = δgauge + δmatter is universaland given by

δmatter = − Nc

8π2λ4topf

4 , δgauge =3f 4(3g4 + 2g2g′2 + g′4)

512π2, (2.46)

with Mtop ≡ λtopv.In the most general case both γgauge and γmatter are quadratically sensitive

to high energy scales, namely if the potential is expressed as in eq.(2.33) asan integral in momentum space

γ ∼∫q dq

(1 +O(q−2) + . . .

). (2.47)

At the same time βgauge and βmatter assume the following behavior

β ∼∫q dq

(q−2 + . . .

), (2.48)

revealing a logarithmic sensitivity to the cutoff of the theory. To improvecalculability and relax this UV sensitivity, therefore keeping under control thefine tuning, generalized sum rules have been studied [45,46]: if the parametersof the theory satisfy these sum rules the divergent behaviors are canceled;they have been derived in close analogy with the Weinberg sum rules forQCD.

In section 3.2 we will comment on the UV sensitivity of the SUSY cases.SUSY introduces contributions to the Higgs potential of opposite spin withrespect to the ones discussed here, but a generalization of the procedure toget eq.(2.33) is straightforward: matter fermions are properly included inchiral superfields, while vectors are part of vector multiplets; also additional

3A numerical analysis confirms this result and shows that typically βgauge is at leastone order of magnitude smaller than βmatter.


chiral superfields will contribute to the gauge part of the potential. We an-ticipate now that only gaugino soft masses can contribute quadratically tothe (negative) Higgs square mass. In the language of eq.(2.47) and eq.(2.48)this result can be seen as the fact that for the matter contribution the mostdivergent terms, quadratically and logarithmically divergent in γmatter andβmatter respectively, are zero because the parameters satisfy a certain condi-tion, while the logarithmic part in γmatter is canceled by the sum of the effectof bosonic and fermionic degrees of freedom: at the end the Higgs potentialis UV finite. In the gauge sector SUSY cancellations are not exact and thereis a leftover dependence on the cutoff of the theory, as we will see in greaterdetail in the next chapter.

2.4.1 Non-SUSY Higgs Mass Estimates

Before analyzing the Higgs potential in SUSY CHM, it might be useful toquickly review the situation in the purely non-SUSY bottom-up construc-tions. We focus in what follows on models where the composite fields are inthe fundamental representation of SO(5). Higher representations lead to amultitude of other fields, they are more complicated to embed in a UV modeland they worsen the problem of Landau poles. Moreover they might lead todangerous tree-level Higgs mediated flavor changing neutral currents [47].It should however be emphasized that they can be useful and can result inqualitatively different results, see e.g. [31] for a recent discussion of the Higgsmass estimate for CHM with composite fermions in the 14 of SO(5). Gener-ically, the Higgs mass is not calculable in CHM, since both γ and β definedin eq.(2.37) are divergent and require a counterterm. The situation improvesif a symmetry, such as collective breaking [30, 48], is advocated to protectthese quantities, at least at one-loop level, or if one assumes that γ and βare dominated by the lightest set of resonances in the composite sector, sat-urating generalized Weinberg sum rules [45, 46]. As far as the Higgs massis concerned, we see from eq.(2.43) that, at fixed ξ, it is enough to make βfinite to be able to predict the Higgs mass.

In CHM with partial compositeness, the largest source of explicit breakingof the global symmetry comes from the mass term mixing the top with thecomposite sector. In first approximation, we can switch off all other sourcesof breaking, including the electroweak SM couplings g and g′. The estimateof the Higgs mass is then necessarily linked to the mechanism generating amass for the top. Let us first consider the case in which both tL and tR are


elementary and mix with their partners. In this case two mass mixing termsεL and εR are required to mix them with fermion states of the compositesector. The top mass goes like

Mtop ∼εLεRMf

sh , (2.49)

where by Mf we denote the mass (taken equal for simplicity) of the lightestfermion resonances in the composite sector that couple to tR and tL. Inthe limit in which ε are the only source of explicit symmetry breaking, asimple NDA estimate gives the form of the factors γ, β entering in the Higgspotential (2.36):4

γ ∼ Nc

16π2ε2M2

f , β ∼ Nc

16π2ε4 . (2.50)

Plugging eqs.(2.50) and (2.49) in eq.(2.43) gives

M2H '

Ncε4

2π2f 2ξ ∼ Nc

2π2M2

top

M2f

f 2(tR elementary) . (2.51)

This estimate reveals a growth of the Higgs mass with the top partners massscale. If one assumes that the composite sector is characterized by the singlecoupling constant gρ [49], we expect that Mf ' gρf . Indirect bounds on the Sparameter require gρf & 2 TeV. For values of f . 1 TeV this implies gρ & 2.In many explicit models [45, 46, 50, 51] it has been shown that such a choiceresults in a too heavy Higgs. Indeed, a 126 GeV Higgs is attained onlyif one assumes that another mass scale characterizes the composite sectorand one has relatively light fermion resonances in the composite sector, withMf < gρf . Although the splitting required between Mf and gρf is modest,it is not easy to argue how it might appear in genuinely strongly couplednon-SUSY theories.

Another possibility is having tL elementary and tR fully composite. Thismeans that the right top is not present among the elementary fields andin the meanwhile in the composite sector there is one state, with the rightquantum numbers, that we identify with tR

5. We can now have a directmixing between tL and tR, in principle with no need of composite massive

4The estimate (2.50) changes when fields in higher representations are considered. For

instance, β ∼ NcM2f ε

2

16π2 when fields in the 14 are considered [31].5Gauge anomalies have to cancel non trivially, that is non independently, among ele-

mentary and composite fermions.


resonances, that can all be taken heavy. Denoting by ε this mass mixingterm, we get

Mtop ' εsh . (2.52)

Proceeding as before, we get

M2H '

Ncε4

2π2f 2ξ =

Nc

2π2M2

top

M2top

v2(tR composite) . (2.53)

We see that the Higgs mass is at leading order independent of the details ofthe composite sector and tends to be too light.6 Of course, this is the case inthe assumption that the top mixing term is the dominant source of explicitSO(5) breaking. One can always add extra breaking terms to raise the Higgsmass. Clearly, this is quite ad hoc, unless these terms are already presentfor other reasons. This happens in the concrete model with composite tRintroduced in [1], where anomaly cancellation and absence of massless non-SM states require adding exotic elementary states that necessarily introducean extra source of explicit SO(5) breaking. We have explicitly verified in themodel of [1] that the estimate (2.53) captures to a good accuracy the topcontribution to the Higgs mass. This is still too light, despite the presenceof additional sources of SO(5) breaking, that cannot be taken too large forconsistency. We conclude that models with a composite tR, at least thosewhere the top sector plays a key role in the EWSB pattern, lead to a toosmall Higgs mass.

Let us now briefly mention on how ξ can be tuned to the desired value.There are essentially two ways to do that in a calculable manner: either|γmatter| � |γgauge|, in which case the cancellation takes place mostly insidethe matter sector, or |γmatter| ∼ |γgauge|, so that the gauge and matter con-tributions can be tuned against each other (see, for example, the discussionin Section 4 of [45]). Both options are generally possible, with the exceptionof minimal (i.e. where one ε is the only source of SO(5) violation in thematter sector) models with a fully composite tR embedded in a fundamentalof SO(5), where one can rely only on the second option.

6 The problem of a too light Higgs when tR is fully composite (when embedded in a5 of SO(5)) was already pointed out in [45], where a formula like eq.(2.53) (see eq.(5.14))was derived for a particular model.


2.5 Experimental probes of Composite Higgs

Models

2.5.1 Direct Searches

In this section we discuss the experimental consequences of pNGB CHM. Thepresence of new, still unobserved, massive states characterizes many modelsof BSM physics: in the case at hand heavy resonances are introduced aspartners for SM fields. The discussion on the Higgs potential in section 2.4outlined that the most relevant ones are top partners because their presence,and their relative “lightness”, is linked to top and Higgs masses. They arecolored fermions transforming in representations of SO(5) × U(1)X wherenow the X charge is not negligible: as we explained a common choice isto obtain the SM hypercharge as Y = T 3

R + X where T 3R is a generator of

SU(2)R ⊂ SO(4) ⊂ SO(5). Therefore under SM gauge group we decompose52/3 = (4 + 1)2/3 = 27/6 + 21/6 + 12/3. We thus expect massive vectorlikequarks and new fermions with exotic electric charge Q = 5/3. The relevanceof searches for resonances has been stressed in [52] where simplified modelshave been introduced; dedicated studies of CMS collaboration [13], along thelines of [53], put a limit of 800 GeV for masses of Q = 5/3 colored fermions at95% c.l. using 19.5 fb−1 collected at

√s = 8 TeV at LHC. The strategy is to

look for top partners pair produced through colored interactions each of themdecaying to a W boson and a top which decays to another W and a bottom :they analyze events with two same sign leptons coming from leptonic decaysof the two (same sign) W bosons. While this bound applies universallyto Q = 5/3 top partners the single production case is model dependent:the cross section for the single production is expected to overcome the pairproduction for masses of the resonance roughly around the TeV. The processinvolves the vertex interaction among an EW vector boson, a quark and oneof its partners, therefore the model dependence rely on the strenght of thevarious mixings εr.

For Q = 2/3 top partners CMS published the results of an inclusive searchbased on 19.5 fb−1 at 8 TeV [54]. They can decay to top and Higgs, top andZ boson or bottom and W boson and masses below 700 GeV are excluced:the exact value depends on the various branching ratios.

Higher dimensional SO(5) representations contain other exotic partners,like the 14 where a fermion with Q = 8/3 is present. The case is studied


in [55], where events with same sign dileptons are analyzed and a bound isput at a mass 940 GeV. Q = 8/3 partners do not couple directly to SM fieldsat the renormalizable level due to their large electric charge: their decay ismediated by a vertex with two W bosons and a top (it can be seen as a twostep decay through a Q = 5/3 fermion). In [55] they also explored the morerealistic scenario with all the states in the 14 roughly at the same mass,obtaining a comparable bound.

While fermion resonances are an essential ingredient of partial compos-iteness, vector resonances are not strictly necessary, even though they havereceived a lot of attention, for instance in [56]. Partners of Z and W bosonswith masses below 1.5 TeV are excluded [57, 58], [59] for limits not fromthe experimental collaborations. While experimental collaborations focusedon benchmark models, in [60] has been considered the more realistic case ofa heavy spin-1 resonance accompanied by fermionic resonances, specificallythose typically arising from fundamental representation of SO(5).

On general grounds one could also expect the presence of gluon reso-nances, massive vectors in the adjoint of SU(3)c: this is particularly true inextra dimensional models where they are identified with KK excitations of theordinary SM gluons, while in pure four dimensional theories they are not un-avoidable. They affect the two point functions of other fermionic resonancesand they enter the Higgs potential at two loops, nevertheless their effect canbe not negligible as long as they do not completely decouple: as stressedin [61] masses above the experimental limit, set by CMS at 2.5 TeV [62], cancontribute to lower the Higgs physical mass by a few percentage points.

2.5.2 Indirect Measurements

Higgs Couplings

Direct detection of a resonance would surely be a reliable sign of BSM physicsand it would also helps in discriminating among several BSM hypothesis.Other indirect indications might come from deviations in the measure of theHiggs couplings: in this case different models may results in comparable pre-dictions and further experiments would be needed. In CHM such deviationsare intimately related to the pNGB nature of the Higgs: couplings to fermionsand EW gauge bosons are dictated by symmetry considerations. For instance


for the coset SO(5)/SO(4) the Lagrangian for gauge bosons contains

1

2

f 2

4s2h[g′2BµBν − 2g′gBµW

3ν + g2(W 1

µW1ν +W 2

µW2ν +W 3

µW3ν )]ηµν (2.54)

coming from the covariant derivative of the Higgs field. The identificationv2 = f 2〈s2

h〉 = f 2ξ yields to the correct values for W and Z masses andensures the relation mW = cos θWmZ . If we expand around the VEV 〈h〉+hfor small fluctuations we find

f 2s2h ' v2 + 2v

√1− ξh+ (1− 2ξ)h2 , (2.55)

whereas the SM is recovered in the limit ξ → 0 at fixed v. Therefore thecoupling V V h and V V hh are modified by a factor

√1− ξ and 1−2ξ respec-

tively. The couplings of the Higgs to the EW bosons are reduced comparedto the SM case in a variety of cosets: an enhancement is obtained in noncompact cosets, like SO(4, 1)/SO(4) [63,64].

The couplings to SM fermions depend on the SO(5) representations ofthe fermion partners. If we choose the fundamental 5 for both left and rightthe mass term is proportional to

f sin2h

f' 2v

√1− ξ + 2h(1− 2ξ) = 2v

√1− ξ

(1 +

h

v

(1− 2ξ)√1− ξ

)(2.56)

and therefore the Higgs coupling to fermions hff with respect to the SMcase is modified by the factor (1− 2ξ)/

√1− ξ.

Couplings to gluons and photons are induced by loops of charged and col-ored particles and are model dependent. The quickest way to compute themis through the low energy theorem [65, 66] which states that an amplitudeinvolving a soft Higgs as external leg can be computed as a function of thesame amplitude without the Higgs, namely

limph→0A(X + h) = h

∂

∂vA(X) (2.57)

if MH is much smaller than the masses in the loop. This is because insertinga Higgs in a propagator means the following replacement in the amplitudeA(X):

1

/q −m→ 1

/q −mh

1

/q −m= h

∂

∂m

1

/q −m. (2.58)


The Jacobian ∂m∂v

contains the Yukawa coupling of the vertex. In fact inter-actions with the Higgs may be generated in a Lagrangian, with the kineticterm consistently neglected in the soft limit, with the formal substitution

m→ m

(1 + g

h

v

)(2.59)

where g is a proper coefficient (g = 1 in the SM); with different values forg the theorem can be generalized to new physics models. As an example ofthe use of eq.(2.57) we compute the coupling of the Higgs to two gluons inthe SM mediated by a top loop. We start with the term in the action

L ⊇ − 1

4g23

F aµνF

µνa (2.60)

with g3 the coupling of SU(3)c such that β 1

g23

= b8π2 . We recall that

∂ 1g23

∂ logMtop

= −2

3

1

8π2(2.61)

due to the changing of the beta function at energies crossing Mtop. With theidentity ∂

∂v= 1

v∂

∂ logMtopand putting everything together we obtain

L ⊇ g23

48π2F aµνF

µνah

v(2.62)

where we have reabsorbed the gauge coupling constant in the vector fields,Aaµ → g3A

aµ; it agrees with the full one-loop computation up to terms of order

O(M2H

M2top

). Notice that the effect does not depend on the top mass, as long

as Mtop � MH because the coupling is proportional to the Yukawa of thetop: in other words a massive field does not decouple because the heavierit is the more it interacts. The vertex with two gluons and n external softHiggs legs is obtained iterating eq.(2.57) on eq.(2.62) and dividing for n! to

avoid multiple counting. Noticing that(∂∂v

)n−1v−1 = (−1)n−1(n−1)!v−n the

resummation of all the vertices hngg results in

L ⊇ g23

48π2F aµνF

µνa log

(1 +

h

v

). (2.63)


More in general the result can be expressed in terms of log det |M(h)|2 and theHiggs dependence is obtained performing the replacement eq.(2.59), whereM(h) is the mass matrix of the fields entering the loop. This holds notonly for the SM but has wider applicability. In [67] the couplings of theHiggs to gluons has been studied extensively taking into account both thecontribution from heavy resonances and the non linearities coming from thecoset structure.

So far all the data extracted from the experimental collaborations at LHCare in good agreement with the expectations from the SM, no significantdeviation for any of the probed couplings of the Higgs to bosons (massivevectors, photons and gluons) or fermions (bottom, tau) is found and theuncertainties shrank toO(10%): see [68–70] and the many references reportedtherein.

Electroweak Precision Observables

In discussing electroweak precision tests (EWPT) we follow the discussionof [71]. New physics effects are included in form factors of EW gauge bosons

ΠV (q2) = ΠV (0) + Π′(0)q2 +1

2Π′′(0)(q2)

2+ . . . (2.64)

where V ∈ {W+W−,W 3W 3, BB,W 3B} as in eq.(2.31). The vector bosonsneed not to be mass eigenstates but simply they are the gauge fields appearingin the covariant derivatives of the SM fields with gauge couplings g0 andg′0: namely they are the elementary fields and in principle they can mixwith heavy vector resonances. Moreover we decide to work with canonicallynormalized fields. For these reasons in the following formulae both the pairsg0, g

′0 and g, g′ appear. While g0 and g′0 are the couplings associated to the

gauging the couplings g and g′ are the observed ones, taking into accountpossible mixings. An example of the mixing will be given in eq.(4.16).

Stopping at the second order in the expansion five out of the twelveparameters are fixed by phenomenological considerations:

Π′W+W−(0) = −g20

g2, Π′BB(0) = −g

′20

g′2, ΠW+W−(0) = g2

0

v2

2. (2.65)

The masslessness of the photon Aµ = Bµ cos θW + W 3µ sin θW is guaranteed

by the following

ΠW 3W 3(0) = − gg0

g′g′0ΠW 3B(0) , ΠBB(0) = −g

′g′0gg0

ΠW 3B(0) , (2.66)


Consequently we are left with the following:

{1g2S = − 1

g0g′0Π′W 3B(0)

2gg′M2

WX = − 1

g0g′0Π′′W 3B(0)

,

{2

g2M2WW = − 1

g20Π′′W 3W 3(0)

2g′2M2

WY = − 1

g′20Π′′BB(0)

,

1g2M2

W T = − 1g20

(ΠW 3W 3(0)− ΠW+W−(0))

− 1g2U = − 1

g20(Π′W 3W 3(0)− Π′W+W−(0))

2g2M2

WV = − 1

g20(Π′′W 3W 3(0)− Π′′W+W−(0))

.

(2.67)

We have distinguished them into three classes: T , U and V vanish ina theory preserving either custodial symmetry or SU(2)L. S and X arenot protected by the custodial symmetry while W and Y are not protectedneither by the custodial nor by the SU(2)L symmetry.

They are related to the S, T, U parameters [72,73]:

S =4 sin2 θW

αS , T =

T

α, U = −4 sin2 θW

αU . (2.68)

where we use α = e2

4π, e = g/ sin θW . In an expansion around q2 = 0 of the

form factors ΠV we expect the coefficient of the n-th term to be suppressedwith respect to the (n− 1)-th one by a factor (MW/Λ)2, where Λ is the scaleat which new physics enters into the form factors. Therefore we can restrictourselves to the set of parameters S, T , Y,W . In pNGB CHM a suppressionby a factor (g/gm)2 is reasonable: thus the most relevant parameters for the

present discussion are S and T .We are interested in contributions to form factors, and hence to S and

T , different from the SM ones and possibly due to new physics. Let usstart considering S. A first important contribution to the ΠW 3B(q2) fromthe strong sector can be interpreted, in the strongly interacting limit, as theexchange of heavy narrow spin-1 resonances: due to partial compositenessthey mix at tree-level with the EW gauge bosons. For example a heavyvector boson ρ in the 6 of SO(4) can be treated with a simplified Lagrangianincluding its coupling constant gm and its mass gmf : the mixing with theSM bosons is proportional to gf . Its effect is

∆Sρ =g2

g2m

ξ . (2.69)


At the loop level the form factors get more contributions from the strongsector: in the narrow resonances approximation other states enter the loop.In particular fermionic partners of the SM fermions, depending on their SO(4)properties, can affect the result. In [74, 75] a one-loop computation hasbeen performed taking into account the non linearities of the couplings. Inthe presence of a ψ4 and a ψ1 resonances respectively in the 4 and 1 ofSO(4) ⊆ SO(5) a non derivative coupling with the Higgs is induced by

ψ4 /D ψ1 + h.c. (2.70)

where Dµ is defined in eq.(2.8). The induced ∆Sf has no definite sign andit can vanish for a particular choice of the coefficient of the above operator:this corresponds to points with enlarged symmetries.

A third important effect is related to the non linearities of the Higgsboson [76]: as we discussed they translate into deviations of the couplings ofthe Higgs to EW vector bosons with respect to the SM values, reobtainablein the limit ξ → 0 at fixed v. In the SM, being a renormalizable theory, Shas to be finite because we cannot add counterterms for it; on the other handthe effective theory at ξ 6= 0 is no longer renormalizable, a UV completion isgiven by the addition of the strongly interacting sector. Therefore we expectS to be sensitive to the cutoff scale, Λ. Notice that, conversely to whathappened before, this contribution does not depend on the details of theUV theory, namely masses and couplings of heavy resonances, but it is fullyunderstood in the effective theory: in this sense it is a IR contribution. Thetotal one-loop computation with the vectors, the Goldstones and the neutralHiggs with modified couplings results in [76]

∆SH =1

6π

g2

16πξ log

Λ

MH

. (2.71)

For what concern T there is no tree-level contibution because the com-posite sector respects a custodial symmetry. At the loop level this symmetryis spoiled therefore we expect a non vanishing ∆T : in particular the contri-bution from fermionic resonances can be estimated to be [30,49]

∆Tferm ∼Nc

16π2

ε4rf2

M2L

ξ . (2.72)

Higgs non linearities produces an IR effect similar to the one in eq.(2.71)

∆TH = − 3g2

32π2 cos2 θWξ log

Λ

MH

. (2.73)


The numerical values for S and T , although in terms of the non hattedquantities S, T and U , can be found in [77,78].

As pointed out in [79,80] models with 5 or 10 representations for Or arefavoured with respect to the minimal models with fermions embedded in thespinorial 4. While corrections to T vanish at tree-level if the BSM physicsrespects a custodial symmetry, one can also include an additional symmetryPLR exchanging L ↔ R, on top of SU(2)L × SU(2)R. Since PLR ∈ O(4)and PLR 6∈ SO(4)7 it can be forced enlarging to the breaking O(5) → O(4)which is equivalent at the level of the algebra but does not allow for spinorialrepresentations. At the same time the former representations automaticallyguarantee that no deviations occurs at tree-level in the coupling of the bottomleft to the Z boson with respect to the SM value: the bound from [81,82] ondeviations is about 10−3. The couplings of top to Z and of top and bottom toW are instead modified but they are less constrained experimentally. Noticethat the requirements studied in [79] to protect the coupling Zbb cannot besatisfied if one wants to embed in (different) SO(5) spurions the top and thebottom quarks: therefore with the most common choice, the one we adoptin the following, a tree-level correction appears proportional to the mixingsεbL of the bottom quark.

7PLR acting on the fundamental 5 of O(5) can be written as diag(−1,−1,−1, 1, 1).

Chapter 3

Supersymmetric CompositeHiggs Models

3.1 General Setup

We now proceed to generalize the pNGB Higgs paradigm in SUSY theories:our models consist of an elementary sector, containing SM fermions, gaugebosons and their supersymmetric partners, coupled to a composite sectorwhere both the global symmetry G and SUSY are spontaneously broken.On top of this structure, in order to have sizable SM soft mass terms, weneed to assume the existence of a further sector which is responsible foran additional source of SUSY breaking and its mediation to the other twosectors. We do not specify it and we parametrize its effects by adding softterms in both the elementary and the composite sectors. Our key assumptionis that the soft masses in the composite sector are G invariant. See fig.3.1 fora schematic representation. The main sources of explicit breaking of G arethe couplings between the elementary and the composite sectors, namely theSM gauge couplings and the top mass mixing terms. We assume that partialcompositeness in the matter sector is realized through a superpotential portalof the form

W ⊃ ε ξSMNcomp , (3.1)

the supersymmetric generalization of eq.(2.18). In eq.(3.1) Ncomp are chiralfields in the composite sector and ξSM denote the SM matter chiral fields.No Higgs chiral fields are present in the elementary sector, since the Higgsarises from the composite sector. The term (3.1) is the only superpotential

40

CHAPTER 3. SUPERSYMMETRIC COMPOSITE HIGGS MODELS 41

term involving SM matter fields. For concreteness, we consider only theminimal custodially invariant SO(5) → SO(4) symmetry breaking patternwith Ncomp in the fundamental representation of SO(5). Like in non-SUSYCHM, the SM Yukawa couplings arise from the more fundamental proto-Yukawa couplings of the form (3.1).1 We do not consider SM fermions butthe top since they are not expected to play an important role in the EWSBmechanism. They can get a mass via partial compositeness through theportal (3.1), like the top quark, or by irrelevant deformations, for instanceby adding quartic superpotential terms.

As we are going to show, the SUSY models we consider can be seen as theweakly coupled description of some IR phase of a strongly coupled theory, inwhich case the Higgs is really composite, or alternatively one can take themas linear UV completions, in which case no compositeness occurs. Dependingon the different point of view, general considerations can be made. If we wantto take our models as UV completions on their own, we might want to extendthe range of validity of the theory up to high scales, ideally up to the GUTor Planck scale. In this setting, introducing gauge fields in addition to theSM gauge fields is disfavoured, because the multiplicity of the involved fieldswould typically imply that the associated gauge couplings are not UV freeand develop a Landau pole at relatively low energies. Avoiding analogousLandau poles for certain Yukawa couplings in the superpotential implies thatthe “composite sector” should be as weakly coupled as possible. However,reproducing the correct top mass forces some coupling to be sizable; in ourexplicit example a Landau pole is reached at a scale around 102f . Viceversa,additional gauge fields are generally required if we assume that the linearmodels considered are an effective IR description of a more fundamentalstrongly coupled theory, like in ref. [1]. We might now assume that the theorybecomes strongly coupled at relatively low scales, such as Λ = 4πf . We canactually determine the low energy non-SM Yukawa and gauge couplings bydemanding that they all become strong around the same scale Λ. As we willsee, light fermion top partners still appear in both cases.

In light of these two different perspectives, we will consider in greaterdetail two benchmark models, with and without vector resonances.

1In the field basis where we remove non-derivative interactions of the pNGB Higgs fromthe composite sector, the Higgs appears in eq.(3.1).


Composite sectorElementary sector

gSM

�

Soft SUSY terms

Figure 3.1: Schematic representation of the structure of our models.

3.2 General Features of the Higgs potential

We now move to realizations of pNGB CHM in concrete models which enjoy arigid N = 1 SUSY. First we consider more specifically the Higgs potential inSUSY models. No tree-level D-term contribution to the potential is presentin our models, in contrast to many SUSY little Higgs constructions. We makecomparisons with little Higgs models [83,84] because they are a realization ofthe idea of CHM different from the one we are pursuing and because they alsohave been considered in SUSY versions [85–89], therefore it is instructive topoint out similarities and differences. The latter are based on global unitarysymmetries, where one typically embeds the two MSSM Higgs doublets intwo distinct multiplets of the underlying global symmetry group. Because ofthat, one generally ends up in having too large D-term contributions to theHiggs mass, whose cancellation usually requires some more model buildingeffort. In our case, instead, the two Higgs doublets are embedded in a singlechiral field q4 that is in the 4 of the unbroken SO(4) group. More precisely,the two Higgs doublets Hu,d are embedded in q4 as follows:

q4 =1√2

(− i(H(u)

u +H(d)d ), H(u)

u −H(d)d , i(H(d)

u −H(u)d ), H

(u)d +H(d)

u

),

(3.2)


where the superscript denotes the up or down component of the doublet.Thanks to the underlying global symmetry, the Hu and Hd soft mass termsare equal, thus vu = vd and tan β = 1. The mass eigenstates are simply thereal and imaginary components of q4. Im q4 is identified with the heavy Higgsdoublet, while Re q4 is the light (SM) Higgs doublet. No D-term contributionaffects the Higgs mass. In fact, at tree-level the SM Higgs is massless andits VEV is undetermined. Of course, the situation changes at one-loop level,because of the various sources of violation of the SO(5) global symmetry.The SM Higgs will still sit along the flat direction (i.e. tan β remains one atthe quantum level), but quantum corrections will lift the flat direction, fixits VEV and give it a mass. As explained at the beginning of section 2.4,being the light Higgs doublet a pNGB, it is convenient to parametrize itspotential in terms of the sine of the field, as in eq.(2.34). From now on, forsimplicity, we denote the SM light Higgs doublet as the Higgs and denoteby h the Higgs field in the unitary gauge, matching the notation with thatintroduced at the beginning of section 2.4.

In the Dimensional Reduction (DRED) scheme the one-loop Higgs po-tential V (1) is given by

V (1)(sh) =1

16π2

∑n

(−1)2sn

4(2sn + 1)mn(sh)

4

(log

m2n(sh)

Q2− 3

2

)=

1

64π2STr

[M4(sh)

(log

M2(sh)

Q2− 3

2

)],

(3.3)

where m2n(sh) are the Higgs-dependent mass squared eigenvalues for the

scalars, fermions and gauge fields in the theory and we have denoted thesliding scale by Q. When the mass eigenvalues are not analytically available,we compute the logM2 term by using the following identity, valid for anarbitrary semi-positive definite matrix M , see e.g. [90]:

M4 logM2 = limΛ→∞

(1

2Λ4 − Λ2M2 +M4 log Λ2 − 2

∫ Λ

0

x5dx

x2 +M2

). (3.4)

The RG-invariance of the scalar potential at one-loop level reads

∂

∂ logQV (1) + βλI

∂

∂λIV (0) − γnΦn

∂

∂Φn

V (0) = 0, (3.5)

where the indices I and n run over all the masses and couplings (includingsoft terms) and all the scalar fields in the theory, respectively, and V (0)


denotes the tree-level scalar potential with the addition of soft terms. Byexpanding eq. (3.3) for sh � 1, we get the explicit form for γ and β definedin eq.(2.37). As we already pointed out in section 2.4, in first approximationwe can switch off all SM gauge interactions and keep only the top mixingmasses ε as explicit source of symmetry breaking. In this limit, only coloredfermion and scalar fields contribute to the Higgs potential (3.3).

When all sources of SUSY breaking, denoted collectively by m, are switchedoff, SUSY requires

limm→0

V (h) = 0 . (3.6)

However, one has to be careful in properly taking the two limits m→ 0, andsh → 0, since in general they do not commute. The cancellation (3.6) is onlymanifest when we first take the m → 0 limit. In practice, however, we onlyexpand in sh since the sources of SUSY breaking cannot be taken too small.

When the soft terms in the composite sector are SO(5) invariant and theSM gauge interactions are switched off the only SO(5) violating term is thesuperpotential term

W ⊇ εLξLOtL + εRξROtR , (3.7)

a superfield generalization of eq.(2.18). It contains the partial compositenessmass mixings among SM fermions and partners as well as an analogous terminvolving scalar fields, as dictated by SUSY. Also we point out that in theungauged SM limit the β-functions βλI and the anomalous dimensions γnappearing in eq.(3.5) are necessarily SO(5) invariant at one-loop level. As aconsequence, neither the second nor the third term in eq. (3.5) can depend onsh and hence the sh-dependent one-loop potential V (1) is RG invariant andfinite. In this case, in contrast to the MSSM, the electroweak scale ξ definedin eq.(2.38) is only logarithmically sensitive to the soft masses when theseare taken parametrically large. A similar structure is found for the one-looppotential in SUSY little Higgs models, where a phenomenon named doubleprotection is said to be at work [91,92]. The global symmetry breaking scalef is quadratically sensitive to the soft mass terms associated to the fieldsresponsible for this breaking when these are taken parametrically large. Inour models such fields are always in the gauge sector, where we provide adynamical mechanism of SUSY breaking.

When the SM gauge interactions are switched on, βλI and γn are no longerSO(5) invariant and can depend on sh. Although holomorphy protects the su-perpotential from quantum corrections, the Kahler potential is renormalized


and the gauging of SU(2)L×U(1)Y explicitly breaks the SO(5) global symme-try. This implies that the physical, rather than holomorphic, couplings of thecomposite sector entering in the superpotential split into several componentswith different RG evolutions, depending on the SU(2)L × U(1)Y quantumnumbers of the involved fields. We discuss this aspect in more detail in ap-pendix A.1. In what follows, we take the physical couplings to be all equalat the scale f . mψ introduced in eq.(3.12) and h, introduced in eq.(3.12) aswell as in eq.(4.3),2 are the couplings whose running is relevant for the Higgspotential.

Similarly, the RG flow induced by the SM gauge couplings gives rise toSO(5) violating contributions to the soft mass terms. In the models we willconsider this dependence appears only at order s2

h. It implies that the RGflow of the tree-level soft terms contributes to γ and induces a quadraticsensitivity to the wino and bino soft terms suppressed by a one-loop factor∼ g2/(16π2). A “Higgs soft mass term” of the form 1

2m2Hf

2s2h, even if absent

at tree-level, is radiatively generated by the bino and wino masses mg. Aradiatively stable assumption about the Higgs soft term m2

H is to take it atthe scale f of order

|m2H | ∼

g2

16π2m2g . (3.8)

In this way, we can neglect its effect on the one-loop potential. Converselythis does not happen for the soft masses of squarks or other scalar fields. Aterm of the form (see [93] and references therein or [94])

g′2Tr[Yim2i ] = g′2S , (3.9)

present in the beta function of the Higgs soft mass, with i spanning overall relevant scalar fields identically vanishes at leading order. In fact softmasses from the composite sector are assumed to be SO(5) invariant, there-fore they do not contribute to the SO(4) breaking soft Higgs mass. For whatconcerns soft masses for elementary scalar fields they do not enter the gaugecontribution at one-loop level: this is a consequence of the fact that the hy-percharge D term does not introduce quartic interactions among the Higgsand elementary fields, because for the models presented

Tr[(U〈q〉)tT 3

R(U〈q〉)]≡ 0 . (3.10)

2Throughout the text we denote also the field associated to the physical Higgs particlewith the same letter h. The difference should be clear from the context and no confusionshould arise.


where the matrix U contains the pNGB Higgs, see eq.(3.17) and the relativediscussion. This statement can be reprashed in the following equivalent way,thanks to the embedding of the Higgs in a 4 of SO(4) as given explicitly ineq.(3.2). The relation tan β = 1 ensures that the physical Higgs is contained

into the sum H(d)u +H

(u)d : therefore the contribution proportional to S cancels

out in the beta function of the soft mass, while of course it affects the massof the other heavy doublet (although this effects does not enter the Higgspotential at one loop and we neglect it). We conclude that apart from theeffect induced by EWinos a vanishing soft mass for the physical Higgs is aradiatively stable assumption at one-loop level.

3.3 Minimal Model without Vector Resonances

Given the features outlined in the previous sections we move to a specificexample. This allows us to make real predictions, mainly about the spectrumof the theory. A simple supersymmetric pNGB Higgs Model with elementarytL and tR can be constructed using two colored chiral multiplets NL,R in the5 of SO(5), two colored SO(5) singlet fields SL,R, two color-neutral multipletsin the 5, q and ψ, and a complete singlet Z. All these multiplets are necessaryto have a linear realization of the global symmetry breaking SO(5)→ SO(4)without unwanted massless charged states. The superpotential reads

W =∑i=L,R

(εiξaiN

ai +λiSiq

aNai )+mNN

aLN

aR+mSSLSR+W0(Z, q, ψ) , (3.11)

whereW0(Z, q, ψ) = hZ(qaqa − µ2) +mψqaψa . (3.12)

Notice that the gauge sector of the model can be seen as a supersym-metrization of the liner σ-model presented in [76] and in the Appendix Gof [56].

We embed the elementary qL and tR into spurions ξL and ξR in the 5 of


SO(5) SU(3)c U(1)XNL 5 3 −2/3NR 5 3 2/3S 1 3 2/3Sc 1 3 −2/3q 5 1 0ψ 5 1 0Z 1 1 0

Table 3.1: Superfields of the model and their quantum numbers under theSU(3)c × SO(5)× U(1)X ⊇ GSM group.

SO(5) for minimality:3

ξL =1√2

bL−ibLtLitL0

, ξR =

0000tR

. (3.13)

The superpotential eq.(3.12) corresponds to an O’Raifeartaigh model ofSUSY breaking. For µ2 > m2

ψ/(2h2), this model has a SUSY breaking mini-

mum with4

〈qa〉 =f√2δ5a , (3.14)

where

f =

√2µ2 −

m2ψ

h2. (3.15)

The scalar VEV’s of Z and ψa, undetermined at the tree-level, are stabilizedat the origin by a one-loop potential. The symmetry breaking pattern is theminimal

SO(5)× U(1)X → SO(4)× U(1)X , (3.16)

3In order to keep the notation light, we omit in what follows the color properties of thefields, that should be clear from the context.

4With a common abuse of language, we denote with the same symbol a chiral superfieldand its lowest scalar component, since it should be clear from the context the distinctionamong the two.


where SU(2)L×U(1)Y is embedded in SO(4)×U(1)X in the standard fashion.The four NGB’s ha can be described by means of the σ-model matrix as

qa = Uabqb = exp

(i√

2

fhaT a

)ab

qb, (3.17)

where T a are the SO(5)/SO(4) broken generators defined as in the AppendixC and q encodes the non-NGB degrees of freedom of q. In the unitary gaugewe can take ha = (0, 0, 0, h), and the matrix U simplifies to

U =

1 0 0 0 00 1 0 0 00 0 1 0 0

0 0 0√

1− s2h sh

0 0 0 −sh√

1− s2h

. (3.18)

The effect of the SUSY breaking is not felt at tree-level by the coloredfields NL,R, SL,R mixing with the top. We add to the SUSY scalar potentialthe soft terms Vsoft = V E

soft + V Csoft, transmitted from the external SUSY

breaking sector, with

V Esoft = m2

tL|ξL|2 + m2

tR|ξR|2 , V C

soft =∑

φi=NL,R,SL,R

m2i |φi|2 , (3.19)

and soft masses for the elementary gauginos of the SM gauge group, mg,1,2,3.We neglect the smaller soft mass terms radiatively induced by W0 and forsimplicity we have not included B-terms. Let us analyze the tree-level massspectrum of the model. We fix the mass parameter mS = 0, since all thestates remain massive in this limit,5 and take λL = λR = λ, so that thecomposite superpotential enjoys a further Z2 symmetry (exchange of L andR fields), broken only by the mixing with SM fermions. We also assumeall parameters to be real and positive. Before EWSB, the fermion massspectrum in the matter sector is as follows. A linear combination of fermions,to be identified with the top, is clearly massless. The SU(2)L doublet withhypercharge 7/6 contained in NL,R does not mix with other fields and has amass equal to MQ7/6

= mN . The doublet with hypercharge 1/6 mixes with

5We checked that, if taken non-zero, its contribution to the potential does not changequalitatively the conclusions of our analysis.


qL and gets a mass MQ1/6=√m2N + ε2L. Two SU(2)L singlets get a mass

square equal to M2S±

= 1/2(m2N + ε2R + λ2f 2 ±

√(m2

N + ε2R)2 + 2m2Nf

2λ2).The scalar spectrum is analogous, with the addition of a shift given by thesoft masses (3.19). After EWSB, the top mass is

Mtop =εLεRfλsh

√1− s2

h√2√m2N + ε2L

√2ε2R + f 2λ2

=εLεRf

2λ2sh√

1− s2h

2√

2MQ 16

MS+MS−

. (3.20)

The gauge sector contains the SM vector superfields w(0) and b(0) and thechiral superfields qa, ψa and Z. For simplicity, we neglect all soft mass termsin this sector, but the SM gaugino masses. Regarding the fermion spectrum,the SO(4) fourplets qn and ψn (n = 1, 2, 3, 4) get a Dirac mass mψ. A linearcombination of ψ5 and Z, we call it p5, gets a Dirac mass, together with q5,√

2(f 2h2 +m2ψ). The orthogonal combination of ψ5 and Z (χ5) is massless

being the goldstino associated to the spontaneous breaking of SUSY. In thescalar sector, Re qn are identified as the pNGB Higgs, while Im qn get a mass√

2mψ. These two are the mass eigenstates of the two Higgs doublets Hu, Hd

introduced in section 3.2. The partners of ψn and p5 will get the same massas the fermions while the partner of the goldstino χ5 is a pseudo-modulus,whose VEV is undetermined at the tree-level. This field is stabilized at theorigin by a one-loop induced potential. Its detailed mass depends on theratio µ2h2/m2

ψ. In the region defined in the next subsection, its mass is oforder mψh/(2π) ∼ 50 ÷ 70 GeV. The real and imaginary parts of q5 have

masses√

2fh and√

2(f 2h2 +m2ψ), respectively.

Let us discuss the possible values of the parameters of the model. De-manding Mtop to be around 150 GeV at the TeV scale gives a lower bound onthe smallest possible value of the Yukawa coupling λ at the scale f , obtainedby taking εL,R →∞ in eq.(3.20):

λmin(f) & 1.2 . (3.21)

An upper bound on λ is found by looking at its RG running. The relevantbeta functions are reported in appendix A.2. For h� 1, the Yukawa couplingλ is UV free for λ(f) . 0.9 and develops a Landau pole for higher values.Demanding that the pole is at a scale greater than 4πf gives the upperbound:

λmax(f) . 1.7 . (3.22)


Putting all together, we see that the maximum scale for which the model istrustable and weakly coupled is obtained by taking λ = λmin, in which casewe get a Landau pole at around 300f . This limiting value is never reachedin realistic situations, but Landau poles as high as 100f can be obtained.The current bound on the top partner with 5/3 electric charge puts a directlower bound on mN [13]:6

MQ7/6= mN & 800 GeV . (3.23)

Demanding a value for λ as close as possible to the minimum value (3.21), thetop mass (3.20) favours regions in parameter space where tL and tR stronglymix with the composite sector: εL,R � mN .

3.3.1 Higgs Mass and Fine Tuning Estimate

As we have already remarked, when V Csoft is SO(5) invariant, the one-loop

matter contribution to the Higgs potential is RG invariant and finite. Sincethe explicit form of βmatter is quite involved, there is not a simple analyticexpression for the Higgs mass valid in all the parameter space. In particularan expansion for small values of εL,R is never a good approximation because,as explained, these mixings should be taken large.

The region of parameter space which realizes EWSB with ξ = 0.1 andgives MH = 126 GeV is essentialy unique. In most of the parameter spaceγgauge and γmatter are both positive and bigger than βmatter, and no tuning ispossible to obtain the right value of ξ. The only region where γgauge < 0 isfound for mg,mψ . f where, however, the size of γgauge is smaller than thenatural size of γmatter, eq.(2.50). The bound (3.8) forces m2

H to be negligiblysmall. From these considerations we see that γmatter has to be tuned in orderto become smaller than its natural value. The requirement of perturbativityup to Λ = 100f fixes λ(f) ' 1.3. Regarding mN , a lower bound is given byeq.(3.23) while an upper bound is given from the fact that, increasing mN

requires a higher value of εL in order to reproduce Mtop, see eq.(3.20), and, asa consequence, γmatter increases, which is the opposite of what it is necessaryto get ξ. This forces mN ∼ f , near its lower bound. Reproducing Mtop fixes

6This bound can be applied directly only if the lightest top partner is this one withQ = 5/3, in which case it decays in tW+ with BR ' 100%. For lower values of the BRthe bound is weaker. We take a conservative approach and use the bound as a constrainton the mass of this particle.


also εL, εR � f . The Higgs mass is not sensitive to the stop soft masses mtL,R

and its correct value is found for composite soft masses m ∼ 3.5f , taken allequal. Finally, in order to fix ξ = 0.1, mtL and mtR have to be tuned in theregion mtL � mtR ∼ m.

An approximate analytic formula for M2H in this region is obtained by

expanding for λf � mN , m� εL,R, where m is a common universal soft massterm (the last one is a good approximation because MH does not depend onthe stop soft masses). In this limit we get

M2H '

Nc

2π2M2

top

M2top

v2

(5 log

( m2

λ2topf

2

)+4x log

( x

1 + x

)+

1

2−4 log 2

), (3.24)

where x = m2

m2N

. It is immediate to see that for values of m & mN ∼ f ,7 a

126 GeV Higgs is reproduced.Let us briefly discuss the fine tuning. We define it here as the ratio

between the value of ξ we want to achieve and its natural value given by(2.39) in absence of cancellations. This is a crude definition, but it has theadvantage to estimate the actual FT provided by cancellations rather thanthe sensitivity, without the need to worry about possible generic sensitivities.The electroweak scale is determined by eq.(2.39). As argued above most ofthe tuning arises within the matter sector. We can then neglect γgauge anddetermine the expected value of ξ by comparing γmatter and βmatter. We get8

γmatter ∼Nc

8π2λ2topf

2m2 , βmatter ∼Nc

8π2λ4topf

4 . (3.25)

The FT can then be written as9

∆ ∼ m2

f 2

1

ξ, (3.26)

and is always higher than the minimum value 1/ξ. From eq.(3.24) we seethat MH grows with m in the region of interest and hence we expect a linearincrease of ∆ with the Higgs mass.

7We have numerically checked that the range of applicability of eq.(3.24) extends tothe region with mN ∼ f .

8As explained below eq.(3.6), the limits sh → 0 and m → 0 do not commute. As aresult, βmatter in eq.(3.25) does not vanish for m→ 0.

9Another possible source of FT might arise from the origin of the scale f as the can-cellation of the two terms in eq.(3.15). In the region of interest no significant cancellationoccurs and we neglect this effect.


120 122 124 126 MH !GeV"2.0

2.5

3.0!"1 #

Figure 3.2: FT of the minimal model, (in %) as a function of the Higgs massfor ξ ' 0.1. In this scan we fixed λ(f) = 1.29 and h(f) = 0.44, so that both λand h reach a Landau pole at the same scale Λ ∼ 100f , mN = 1.2f and pickedrandomly εL ∈ [8.5f, 10f ], mψ ∈ [0.7f, 1.5f ], mg ∈ [0.5f, f ], m ∈ [2.5f, 4.5f ],mtL ∈ [4.5f, 6.5f ], mtR ∈ [f, 3f ] and m2

H within the bound (3.8). We fixed Mtop

by solving for εR and then selected points with ξ ' 0.1. The pink strip representsthe Higgs mass 1σ-interval as reported in ref. [95].

In order to check these considerations we performed a parameter scan inthe restricted region described at the beginning of the section. We fixed thetop mass by solving for εR and then obtained the minimum of the potentialand the Higgs mass from the full one-loop expression of eq.(3.3). We reportin fig.3.2 a plot of the FT computed using the standard definition of ref. [19]as a function of the Higgs mass. As can be seen, we obtain ∆−1 ∼ 2% forMH = 126 GeV, in reasonable agreement with the rough estimate (3.26).

Let us now discuss the spectrum of new particles. In this region, theelectroweak gauginos are relatively light, mg . f ∼ 800 GeV and the twohiggsino doublets (from ψn and qn) have also a mass mψ ∼ 800 GeV. Thestops and their partners are heavy, above 2 TeV, while the fermion top part-ners are usually below the TeV, the lightest being the singlet with Q = 2/3and a mass MS− ' 660 GeV.10 The gluinos do not contribute to the Higgs

10The recent CMS analysis [54] rules out charge 2/3 top partners below ∼ 700 GeV.A careful phenomenological analysis should be performed to check if the model with the


SO(5) SO(4)2 SU(3)c U(1)XNL 5 1 3 −2/3NR 5 1 3 2/3XL 1 4 3 2/3XR 1 4 3 −2/3q 5 4 1 0Z 1⊕ 14 1 1 0

Table 3.2: Chiral superfield content of the model with vector resonances.

potential at one-loop, therefore they can be taken heavy (above the experi-mental bounds) without increasing the fine tuning.

3.4 Road to Vector Resonances

A modification of the model is obtained introducing partial compositenessalso for EW gauge bosons mixing them with heavy spin-1 resonances. In ourlinear realization this is achieved enlarging the pattern of symmetry breakingfrom SO(5) → SO(4) to SO(5)× SO(4)2 → SO(4)D: the gauging of theadditional SO(4)2 provides the vector resonances and keeps the number ofuneaten Goldstone modes to be identified with the components of the Higgsdoublet equal to four. The field content is reported in table 3.2 while thesuperpotential is

W =∑i=L,R

(εiξaiN

ai +λiX

ni q

anN

ai )+mNN

aLN

aR+mXX

nLX

nR+W0(Z, q) , (3.27)

where NL,R and XL,R are colored fields in the (5,1) and (1,4) of SO(5) ×SO(4)2, respectively, and q is a color-singlet in the (5,4) (a = 1, . . . , 5, n =1 . . . , 4). The spurions ξL,R are taken as in eq.(3.13). The superpotentialterm W0 reads

W0 = h(qnaZabq

nb −

f 2

2Zaa

), (3.28)

where Z is a field in the symmetric 14⊕ 1 of SO(5).

benchmark parameters taken is ruled out or not. Slightly decreasing ξ or the scale of theLandau pole are two possible solutions to increase the mass of MS− beyond 700 GeV.


The extra gauge sector is strongly coupled and it has a Landau pole ata low scale, around 10f . Also, due to the augmented multiplicity of fields,the couplings λ and h enter the strong regime at the same scale. This caseis discussed in [2] in a manner similar to the previous case. Neverthelesshere we emphasize that it can be nicely seen as an effective field theory fora theory valid up to arbitrary high energy in the limit of ungauged SM. Infact the superfields, with the addition of a colored state Y , can be groupedas

q =

(XL, XR

q

), M =

(Y NL,R

N tL,R Z

). (3.29)

The colored fields are grouped embedding the SU(3)c × U(1)X in a largerSO(6) such that 6 = 32/3 + 3−2/3. The superfield q is written as a (6 + 5)× 4matrix while M is a (6 + 5)× (6 + 5) symmetric matrix. The transformationproperties of the field Y are consequently deduced: it transforms as a 6× 6.Also the couplings

√2λL,

√2λR and h can be collected in a new coupling,

which we call again h. The cubic terms of eq.(3.27) and eq.(3.28) can bewritten as

W = hTr[qMq] . (3.30)

In the limit of decoupled SM the SO(6)×SO(5) group is a global symme-try, while the SO(4)2 is gauged. It can be viewed, through Seiberg duality,as the low energy description of a UV free supersymmetric QCD (SQCD)with a SO(11) gauge group and Nf = 11 flavors in the fundamental [1]: thegauge group SO(4)2 is emergent and we refer to it as magnetic, in contrastto SO(11) which we call electric. We recall the details of Seiberg dualityfocusing on orthogonal groups in the next section and we devote the nextchapter to the discussion of this model from the UV completed point of view.

3.5 Seiberg Duality for Orthogonal Groups

In this section we briefly review Seiberg duality for orthogonal groups, [96,97];similar considerations can be made for other Lie groups, although we focuson orthogonal for definiteness and because it is the case of our interest.

We first introduce N = 1 SQCD with SO(N) gauge group and Nf chiralsuperfields Q in the fundamental: they transform in the fundamental of aglobal SU(Nf ). N and Nf play a crucial role in determining the dynamics ofthe theory and for different values we have qualitatively different situations,


N − 2 32 (N − 2) 3(N − 2)

runaway free magnetic phase interacting fixed point free electric phase

UV free IR free

confining non confining

Table 3.3: Diagram for SO(N) SQCD: the number of flavor Nf increasesfrom left to right.

as we are going to recall. The coefficient of the beta function of the gaugecoupling is bel = 3(N − 2) − Nf , therefore the theory is IR free for Nf ≥3(N −2); for Nf < N −4 gaugino condensation occurs, while the cases Nf =N−4, N−3, N−2 have a richer vacuum structure. For 0 < 3(N−2)−Nf � 1the theory is expected to have an interacting fixed point: in fact in this rangeof parameters the one-loop coefficient of the beta function for g is positivewhile the two-loop coefficient is negative, therefore it exists a value of g suchthat the beta function vanishes: it can be shown that g ∼ 1/N . An elegantway to derive the two-loop result is to relate the physical gauge couplingto the holomorphic coupling which, being protected by SUSY, runs only atone-loop and whose beta function is exactly proportional to bel [98]. It canbe argued that the existence of an interacting fixed point extends within aconformal window defined by

3

2(N − 2) < Nf < 3(N − 2) . (3.31)

The position of the lower bounds can be inferred inspecting the scalingdimension of the gauge invariant scalar operator M = QQ: because of anunitarity bound valid in any CFT the dimension of any scalar operator can-not be smaller than 1 [99]. At the same time the current associated to theR symmetry and the stress energy momentum tensor (related to the currentassociated to dilatations) sit in the same real vector superfield. In a supercon-formal field theory (SCFT) both currents are conserved and the associatedcharges are proportional, in particular11

2 dimM = 3R(M) (3.32)

11This holds true for primary chiral operators: for a discussion see [37].


and the unitarity bound becomes R(M) > 2/3. Combining R(M) = 2R(Q)and the expression of R(Q) given in table 3.4 gives the bound

Nf > 3(N − 2)/2 . (3.33)

An electric magnetic duality, named after Seiberg, can be formulated forSO(N) SQCD as long as

N − 2 < Nf < 3(N − 2) . (3.34)

The dual theory is a supersymmetric gauge theory with magnetic quarks inthe fundamental of an SO(Nf −N + 4) gauge group and a neutral meson M :it has a non perturbative superpotential of the form

Wmag = hqMq . (3.35)

Consistently the dual theory has an interacting fixed point in the same con-formal window where both h and the gauge coupling are attracted to fixedvalues. Notably the magnetic theory is IR free for the values N − 2 < Nf ≤32(N − 2). In this region the electric theory is strongly coupled in the IR and

confining, and the magnetic dual can be seen as an effective theory for theIR dynamics, otherwise intractable. On the other hand the magnetic theoryprovides, in terms of weakly coupled degrees of freedom, a description validup to a certain cutoff, above which it becomes inconsistent and it is UV com-pleted by the electric theory; see for instance [100]. Finally for Nf ≤ N − 2the magnetic dual is not defined and the original theory typically develops anon perturbative superpotential or a quantum moduli space of vacua: we donot discuss these cases. We summarize this behavior in table 3.3.

In table 3.4 we recall the superfield content of the electric and the mag-netic theories with the gauged and global symmetry.

The running of the gauge coupling constant in a gauge theory producesa scale which is associated to non perturbative effects, as it is well known:

Λ = µ exp

(− 8π2

bg2(µ)

)(3.36)

In our case of Seiberg duality the requirement that the two coupling con-stants, electric and magnetic, match (up to a phase) at a certain scale µtranslates to the following

Λbelel Λbmag

mag = (−1)Nf−NµNf (3.37)


SO(N)el SU(Nf ) U(1)R

QNI N Nf

(Nf−N+2)

Nf

SO(Nf −N + 4)mag SU(Nf ) U(1)RqI,n Nf −N + 4 Nf

N−2Nf

MIJ 1 12Nf (Nf + 1)

2(Nf−N+2)

Nf

Table 3.4: Field content of the electric and magnetic theories.

The sign is fixed by other considerations, namely consistency of this relationand the duality. Eq.(3.37) tells us that g−2

el + g−2mag ∼ const and in the

conformal window at the IR fixed point the weaker one coupling is, thestronger the other.

In the range N−2 < Nf ≤ 32(N−2) the magnetic theory is IR free and it

can be seen as an effective theory valid up to Λmag, while its UV completion,the corresponding electric theory, becomes non perturbative at and below ascale Λel. It is natural to choose the parameters of the theory such that thereis only one scale

Λ = Λel = Λmag = µ . (3.38)

In the following we will assume this is the case and we will refer to the uniquenon perturbative scale of the theory Λ.

Several checks of this proposed duality have been verified: for instancethey enjoy the same global symmetries, the ’t Hooft anomalies match, andadding a mass term for quarks in a theory results in higgsing part of the gaugegroup of the dual theory and vice versa. The gauge invariant operators ofthe electric and magnetic theory are in one to one correspondence:

♠ QNI Q

NJ

♥ εN1...NNQN1I1. . . QNN

IN

♦ εN1...NN−4W1...W4QN1I1. . . Q

NN−4

IN−4(Wα

elWel,α)W1...W4

♣ εN1...NN−2W1W2QN1I1. . . Q

NN−2

IN−2WW1W2el,α

, (3.39)


♠ MIJ

♥ εI1...INJ1...JNf−N εn1...nNf−Nw1...w4qJ1n1 . . . qJNf−NnNf−N

(WαmagWmag,α

)w1...w4

♦ εI1...IN−4J1...JNf−N+4εn1...nNf−N+4

qJ1n1 . . . qJNf−N+4nNf−N+4

♣ εI1...IN−4J1...JNf−N+2εn1...nNf−N+2w1w2q

J1n1 . . . qJNf−N+2nNf−N+2Ww1w2mag,α

.

(3.40)Notice that the dual quarks bilinear qq in the IR is fixed to zero by the

F term equations implied by the superpotential eq.(3.35).In the regime N − 2 < Nf ≤ 3

2(N − 2) the SUSY gauge theory confines

at a scale non perturbatively generated: below that scale we have a descrip-tion in terms of an effective theory for vectors, scalars and fermions whichare free in the IR but strongly coupled in the vicinity of the cutoff of thetheory. Therefore we use this theory, gauging a subgroup of the global sym-metries, as a candidate for the BSM sector, generically introduced in chapter2, providing a pNGB composite Higgs. An interesting phenomenon occursif a mass term is turned on for the quarks in the electrical theory: near theorigin in moduli space there exists a metastable vacuum in which SUSY isspontaneously broken, to which we refer as the Intriligator Seiberg and Shih(ISS) vacuum [101]. The original computation has been possible only thanksto the Seiberg duality since the vacuum is found in terms of magnetic vari-ables. Since the lifetime can be made parametrically large the vacuum isinteresting also from a phenomenological point of view, as a SUSY break-ing sector. We put ourselves in the ISS vacuum but we do not rely on thisSUSY breaking to generate soft masses: we are concerned with the patternof breaking of bosonic symmetries and, as already mentioned, we appeal toanother unspecified SUSY breaking source.

Chapter 4

A UV Complete Model

4.1 The Basic Construction

As argued in subsection 3.4 the model there presented has a UV completion interms of a confining SQCD theory described, at low energy, through Seibergduality, recalled in section 3.5. Before discussing in detail the model we startthis chapter with a more general approach, namely we do not completely fixthe rank of the flavor group spontaneously broken: in fact the key pointsunderlying our models are best illustrated in a set-up where we keep onlythe essential structure. We then converge to specific models. We focus onconstructions where the Higgs is the NGB of an SO(5)/SO(4) coset, but thegeneralization to other cosets should be obvious.

Consider an N = 1 SUSY SO(N) gauge theory with Nf = N flavors inthe fundamental of SO(N), with superpotential

Wel = mabQaQb + λIJKQ

IQJξK . (4.1)

In the first term of eq.(4.1), we split the flavor index I in two sets I = (i, a),a = 1, . . . , 5, i = 6, . . . , N . The fields ξK are singlets under SO(N) andin general can be in some representation of the flavor group Hf ⊂ Gf leftunbroken by the Yukawa couplings λIJK . The ξK ’s are eventually identifiedas the visible chiral fields, such as the top fields. We take λIJK � 1, sothat these couplings are marginally relevant, with no Landau poles, and canbe considered as a small perturbation in the whole UV range of validity ofthe theory. We assume the presence of an external source of SUSY break-ing, whose origin will not be specified, that produces soft terms for all the

59

CHAPTER 4. A UV COMPLETE MODEL 60

SM gauginos and sfermions. For simplicity, we neglect for the moment thedynamics of the singlets ξK and the impact of the external source of SUSYbreaking in the composite sector. We take the quark mass matrix propor-tional to the identity, mab = mQδab, to maximize the unbroken anomaly-freeglobal group. For λIJK = 0, this is equal to

Gf = SO(5)× SU(N − 5) . (4.2)

We take |mQ| = ε2Λ� Λ, where Λ is the dynamically generated scale of thetheory.

For N ≤ 3(N−2)/2, namely N ≥ 6, the theory flows to an IR-free theorywith superpotential [96,97]

Wmag = hqIMIJqJ − µ2Maa + εIJKM

IJξK , (4.3)

whereεIJK = λIJKΛ, µ2 = −mQΛ = (εΛ)2. (4.4)

For simplicity, we identify the dynamically generated scales in the electric andmagnetic theories,1 whose precise relation is anyhow incalculable. The fieldsqI are the dual magnetic quarks in the fundamental representation of the dualSO(Nf − N + 4)m = SO(4)m magnetic gauge group, with coupling gm, andM IJ = QIQJ are neutral mesons, normalized to have canonical dimensionone. The Kahler potential for the mesons M IJ and the dual quarks qI istaken as follows:

K = Tr[M †M ] + q†IeVmagqI , (4.5)

where Vmag is the SO(4)m vector superfield.The original Yukawa couplings λIJKQ

IQJξK in the electric theory flowin the IR to a mixing mass term εIJKM

IJξK between elementary and com-posite fields, the SUSY version of the fermion mixing terms appearing inweakly coupled models with partial compositeness [28]. The quark massterm mQQ

aQa, introduced to break the flavor group from SU(N) down toSO(5)× SU(N − 5), is also responsible for a spontaneous breaking of SUSYby the rank condition, as shown in [101]. Up to global SO(5) × SO(4)mrotations, the non-SUSY, metastable, vacuum is at

〈qnm〉 =µ√hδnm , (4.6)

1We refer to the UV and IR theories as electric and magnetic theories, respectively.


with all other fields vanishing. For simplicity, in the following we take µ andh to be real and positive. In eq.(4.6) we have decomposed the flavor indexa = (m, 5), m,n = 1, 2, 3, 4, and we have explicitly reported the gauge indexn as well. When λIJK = 0, the vacuum (4.6) spontaneously breaks

SO(4)m × SO(5)→ SO(4)D , (4.7)

where SO(4)D is the diagonal subgroup of SO(4)m × SO(4). The groupSU(N − 5) remains untouched, being qna singlet under it. In the global limitgm → 0, this symmety breaking pattern results in 10 NGB’s:

Re (qmn − qnm) : along the broken SO(4)m × SO(4) directions , (4.8)√2 Re qn5 : along the broken SO(5)/SO(4)D directions . (4.9)

For gm 6= 0, the would-be NGB’s (4.8) are eaten by the SO(4)m magneticgauge fields ρµ, that become massive, while the NGB’s (4.9) remain masslessand are identified with the 4 real components of the Higgs field.

The remaining spectrum of the magnetic theory around the vacuum (4.6)is easily obtained by noticing that all fields, but the magnetic quarks qn5 andthe mesons M5n, do not feel at tree-level the SUSY breaking induced by theF -term of M55:

FM55 = −µ2. (4.10)

The chiral multiplets (qmn +qnm)/√

2 and Mmn combine and get a mass 2√hµ,

as well as the multiplets Mim and qmi that form multiplets with mass√

2√hµ.

The chiral multiplets (qmn − qnm)/√

2 combine with the SO(4)m vector multi-

plets to give vector multiplets with mass√

2hgmµ. As we have just seen, the

NGB scalar components Re (qmn − qnm) are eaten by the gauge fields, whileIm (qmn − qnm) get a mass by the SO(4)m D-term potential. Similarly, thefermions (ψqmn −ψqnm)/

√2 become massive by mixing with the gauginos λmn.

The chiral multiplets Mij and Mi5 remain massless.The scalar field M55 is massless at tree-level and its VEV is undetermined

(pseudo-modulus). This is stabilized at the origin by a one-loop inducedColeman-Weinberg potential, as we will shortly see. Its fermion partneris also massless, being the goldstino. Around M55 = 0, the fermions ψq5and ψM5m mix and get a mass

√2√hµ, the scalars M5m get the same mass.

Im qm5 get a mass 2√hµ, while Re qm5 remain massless, the latter being indeed

NGB’s. The fate of M55 is determined by noticing that the superpotential of


the M55 −M5m − qm5 sector is

Wmag ⊃ −µ2M55 +√

2hµqm5 M5m + h(qn5 )2M55 , (4.11)

that is a sum of O’Raifeartaigh models. The associated one-loop potential iswell-known (see e.g. appendices A.2 and A.3 of [101]). The pseudo-modulusM55 is stabilized at zero, and gets a one-loop mass

m2M55

=h4µ2

π2f(h) (4.12)

where f(h) =[

(1+h)2

hlog 1+h

h− (1−h)2

hlog 1−h

h− 2]. The SM vector fields are

introduced by gauging a subgroup of the flavor symmetry group

Hf ⊇ SU(3)c × SU(2)0,L × U(1)0,Y (4.13)

that is left unbroken when we switch on the couplings εIJK . We embed SU(3)cinto SU(N − 5) and SU(2)0,L ×U(1)0,Y in SO(5)×U(1)X , where U(1)X is aU(1) factor coming from SU(N − 5) needed to correctly reproduce the SMfermion hypercharges. The details of the embedding are model-dependentand will be considered in the next sections. We identify SU(2)0,L as thesubgroup of SO(4) ∼= SU(2)0,L × SU(2)0,R ⊂ SO(5). The hypercharge Yis given by Y = T3R + X, where T3R and X are the generators of the σ3

direction U(1)0,R ⊂ SU(2)0,R and of U(1)X , respectively. Denoting by AaLµ(a = 1, 2, 3), A3R

µ and Xµ the SU(2)0,L × U(1)0,R × U(1)X gauge fields andby g0 (the same for SU(2)L and U(1)R, for simplicity) and gX their gaugecouplings, we have (see appendix C for our group-theoretical conventions)

AaLµ = W aµ , A3R

µ = cXBµ , Xµ = sXBµ , (4.14)

where

cX =gX√g2

0 + g2X

=g′0g0

, sX =g0√

g20 + g2

X

. (4.15)

The SU(2)0,L×U(1)0,Y gauge fields W aµ and Bµ introduced in this way are not

yet the actual SM gauge fields, because the flavor-color locking given by theVEV eq.(4.6) generates a mixing between the SO(4)m ∼= SU(2)m,L×SU(2)m,Rmagnetic gauge fields and the elementary gauge fields. This explains thesubscript 0 in SU(2)L,R and U(1)Y,R and in g and g′ in eq.(4.15). The combi-nation of fields along the diagonal SU(2)L×U(1)Y ⊂ SO(4)D×U(1)X group


is finally identified with the SM vector fields. The SM gauge couplings g andg′ are given by

1

g2=

1

g2m

+1

g20

,1

g′2=

1

g2m

+1

g′20. (4.16)

This mixing between elementary and composite gauge fields is analogousto the one advocated in bottom-up 4D constructions of composite Higgsmodels. The situation is simpler for the color group, since the gauge fieldsof SU(3)c are directly identified with the ordinary gluons of QCD; since thegroup SU(N−5) in eq.(4.2) contains SU(3)×U(1), the minimal anomaly-freechoices are SO(6) and SU(4)2.

The set-up above is still unrealistic because of the presence of unwantedexotic massless states (Mij and Mi5). There are various ways to address thesepoints. We do that in the following, where we consider in greater detail thetwo models with SO(6) and SU(4), corresponding to Nf = 5 + 6 = 11 andNf = 5 + 4 = 9 flavors, respectively.

4.2 A Semi-Composite tR

With the general idea presented in the previous section 4.1, we build theexplicit UV completion mentioned in section 3.4, namely we specialize to thecase of a SUSY SO(11) gauge theory with Nf = N = 11 electric quarks. Wealso have two additional singlet fields, Sij and Sia, transforming as (1,20⊕1)and (5,6) of SO(5)×SU(6), respectively.3 We add to the superpotential (4.1)the following terms:

1

2m1SS

2ij + λ1Q

iQjSij +1

2m2SS

2ia + λ2Q

iQaSia . (4.17)

The mass terms in eq.(4.17) break the SU(6) global symmetry to SO(6). Thetotal global symmetry of the model is then

Gf = SO(5)× SO(6) . (4.18)

2The algebras of SO(6) and SU(4) are isomorphic. We embed the SU(3)c such that thefundamental 3 is contained in the fundamental of SO(6) and SU(4), a 6 and a 4.

3See [102] for a similar set-up in the context of models with direct gaugino mediationof SUSY breaking.


For m1S,2S > Λ, the singlets Sij and Sia can be integrated out in the electrictheory. We get4

W effel = mabQ

aQb − λ21

2m1S

(QiQj)2 − λ22

2m2S

(QiQa)2 . (4.19)

In the magnetic dual superpotential, the quartic deformations give rise tomass terms for the mesons Mij and Mi5:

Wmag ⊃ −1

2m1M

2ij −

1

2m2M

2ia , (4.20)

where

mi =Λ2λ2

i

miS

, i = 1, 2 . (4.21)

The mass deformations do not affect the vacuum (4.6), but obviously changethe mass spectrum given in section 4.1. The multiplets Mij and Mi5 are nowmassive, with masses given by m1 and m2, respectively, and the multipletsMim and qmi form massive multiplets with squared masses (m2

2 + 16hµ2 ±m2

√m2

2 + 32hµ2)/8. We take the masses m1 and m2 as free parameters,although phenomenological considerations favour the values of m2 for whichthe mesons Mia, the ones that are going to mix with the elementary SMfields, have a mass around µ. We summarize in table 4.1 the gauge andflavor quantum numbers of the fields appearing in the electric and magnetictheories. We embed SU(3)c into SO(6) and SU(2)0,L × U(1)0,Y in SO(5) ×U(1)X , where U(1)X is a U(1) factor coming from SO(6) (see appendix C).We consider in what follows the top quark only, since this is the relevantfield coupled to the EWSB sector. In terms of the UV theory, we might haveYukawa couplings of the top with the electric quarks, or mixing terms withthe singlet fields. When the singlets are integrated out, we simply get a shiftin the mixing of the top with the meson fields. So, without loss of generality,we can ignore mixing terms between the top and the singlets. The mostgeneral mixing term is then

λL(ξL)iaQiQa + λR(ξR)iaQiQa . (4.22)

4Of course, we could have started directly by deforming the superpotential (4.1) withthe irrelevant operators quartic in the quark fields appearing in eq.(4.19). However wewant to emphasize how easy is to UV complete the above quartic terms. See [103] forstudies of ISS theories deformed by irrelevant operators quartic in the quark fields.


SO(11)el SO(5) SO(6)QNi 11 1 6

QNa 11 5 1

Sij 1 1 20⊕ 1Sia 1 5 6

(a)

SO(4)m SO(5) SO(6)qni 4 1 6qna 4 5 1Mij 1 1 20⊕ 1Mia 1 5 6Mab 1 14⊕ 1 1

(b)

Table 4.1: Quantum numbers under Gf and the strong gauge group of thematter fields appearing in the composite sector of model I: (a) UV electricand (b) IR magnetic theories.

We have written the mixing terms in a formal Gf invariant way in terms ofthe fields ξL and ξR. These are spurion superfields, whose only dynamicalcomponents are the SM doublet superfields QL = (tL, bL)t and the singlet tc,whose θ-component is the conjugate of the right-handed top tR. In order towrite ξL and ξR in terms of qL and tc, we have to choose an embedding ofSU(3) ⊂ SO(6):

(ξL)ia =

b1 −ib1 t1 it1 0−ib1 −b1 −it1 t1 0b2 −ib2 t2 it2 0−ib2 −b2 −it2 t2 0b3 −ib3 t3 it3 0−ib3 −b3 −it3 t3 0

2/3

, (ξR)ia =

0 0 0 0 (tc)1

0 0 0 0 i(tc)1

0 0 0 0 (tc)2

0 0 0 0 i(tc)2

0 0 0 0 (tc)3

0 0 0 0 i(tc)3

−2/3

,

(4.23)in terms of SO(6) × SO(5) multiplets, where the superscript in the fieldsdenote the color SU(3)c index. The subscript ±2/3 denotes the U(1)X chargeof the fermion. The terms in eq.(4.22) explicitly break the global group Gf

of the composite sector and in the magnetic theory they flow to

εL(ξL)iaMia + εR(ξR)iaMia . (4.24)

These are the mixings for partial compositeness of the top, eq.(2.18). We addsoft terms in analogy to eq.(3.19): from a phenomenological point of viewthe only needed soft terms are masses for elementary sparticles and gauginos,because the composite states are all massive. We then add

Vsoft = m2L|tL|

2 + m2R|tR|

2 +(1

2mg,αλαλα + h.c.

), (4.25)


where λα are the SM gauginos and α = 1, 2, 3 runs over the U(1)0,Y , SU(2)0,L

and SU(3)c groups. In order to simplify the expressions below, we take theSM soft terms larger than µ. Due to the terms in eq.(4.24) and the interac-tions with the SM gauginos, the SUSY breaking is transmitted to the compos-ite sector as well. More in detail, the Dirac fermions

(λmn, (ψqmn − ψqnm)/

√2)

mix with the SM gauginos: as a result the former get splitted into two Majo-

rana fermions with masses√

2hgmµ±δmλ. Expanding for heavy SM gauginos,

we have

δmλ,α ∼g2αµ

2 2h

2mg,α

. (4.26)

Similarly, the scalar mesons and magnetic quarks that mix with the stopsget soft terms of order

m2s ∼ −|εL,R|2 , (4.27)

that tend to decrease their SUSY mass value. The spectrum of the fieldsin the Mi5 and in the Mim-qmi sectors is affected by the terms in eq.(4.24),while all the other sectors are unchanged. We see that a linear combinationof fermions given by tR and the appropriate components of ψMia

remainsmassless. This field is identified with the actual SM right-handed top. Asimilar argument applies to tL. At this stage, the “goldstino” ψM55 is stillmassless. In the case in which we also consider soft terms in the electricSO(N) theory, the mesons Mab get a non-vanishing VEV and a mass for ψM55

can be induced from higher dimensional operators in the Kahler potential.Independently of this effect, a linear combination of ψM55 and the goldstinoassociated to the external SUSY breaking is eaten by the gravitino, whilethe orthogonal combination gets a mass at least of order of the gravitinomass (see [104] for an analysis of goldstini in presence of multiple sectors ofSUSY breaking and specifically [105] for a set-up analogous to the one we areadvocating here). We do not further discuss the mechanisms through whichψM55 can get a mass.

More generally SUSY breaking is mediated to both the elementary sectorand the composite one and without introducing further complications weassume it is mediated by the same mechanism, although we do not specifyit: if the mediation scale M is lower than the Seiberg scale Λ the appropriatedescription is in terms of magnetic operators, while if the mediation occursat higher scale we have to use electric operators. In both cases the RGevolution of the SO(11)el SQCD is well captured by Seiberg duality onlyif the SUSY breaking parameters are small compared to Λ: the numerical


analysis we performed showed us that this is always the case. The conditionM < Λ, together with F < M2 where F is the non vanishing SUSY breakingF term of the hidden sector results in unacceptably low soft masses becauseΛ ' 10 TeV, as can be seen from the discussion above, and from the resultswe illustrate in the following sections. Moreover we do not have a cleardescription of such soft terms in terms of UV operators. Therefore we excludethe case of a low scale mediation and we concentrate on the other: we addto the Lagrangian eq.(4.25) the following potential in terms of electric scalarfields

Vsoft,el = m21elQ

†aQa + m22elQ

†iQi = m21el(Q

†aQa + ωQ†iQi) , (4.28)

accompanied with SO(11)el gaugino soft Majorana masses; the last equalitydefines ω = (m2el/m1el)

2. The deformation induced in the magnetic theoryis

Vsoft = m2tL|tL|2 + m2

tR|tR|2 + (

1

2mg,αλαλα + h.c.) + (4.29)

+m21|Mia|2 + m2

2|Mab|2 + m23|qi|2 − m2

4|qa|2 − m25|Mij|2 ,

In section 4.5 we discuss how to derive soft magnetic terms in eq.(4.29) fromeq.(4.28). The result is

m21 =

1

8(3 + 2ω)m2, m2

2 =1

8(11− 6ω)m2, m2

3 =5

16(1− 2ω)m2,

−m24 = − 1

16(11− 6ω)m2 − m2

5 = −5

8(1− 2ω)m2

(4.30)

where m = m1el. Similarly magnetic SO(4)m gauginos acquire a Majoranamass of the form

1

2mλλ

AλA + h.c. (4.31)

and in section 4.5 we derive mλ starting from a mass for SO(11)el gauginos,even though we do not use this expression and we let mλ be a free parameterbecause SO(11)el gaugino mass does not enter any other computation.

The condition ω < 12

ensures the positivity of all squared masses as definedabove. This means we have some tachyonic direction: qna , which are acquiringa vev in any case and Mij which have a nonzero (and possibily “large”)supersymmetric mass m1. Because of this soft terms the vacuum eq.(4.6)gets modified to

〈qmn 〉 =µ√hδmn , where µ =

√µ2 +

m24

2h. (4.32)


Finally in full analogy with eq.(3.17) and eq.(3.18) the matrix describingnonlinearly the Goldstone bosons is extracted from

qnb = exp(i√2

fhaTa +

i√2fπaTa

)bcqmc exp

( i√2fπaTa

)mn, (4.33)

where f = µ√

2h

and the broken generators T a are defined in appendix C.

The broken generators T a collect T aL,R. In the unitary gauge the πa are eatenby the vector bosons; in the unitary gauge also for the EW breaking bosonsthe U matrix becomes exactly of the same form as in eq.(3.18).

4.3 Landau Poles

Similarly to what happens in models with direct gauge mediation of SUSYbreaking, where the SM group is obtained by gauging a global subgroup ofthe hidden sector, one should worry about the possible presence of Landaupoles in the SM couplings, the QCD coupling α3 in particular, due to theproliferation of colored fields. Our model is no exception and Landau polesdevelop for the SM gauge couplings. In order to simplify the RG evolution,we conservatively take all the masses of the magnetic theory to be of orderµ, SM superpartners included, with the exception of the mesons Mij, whosemass m1 is determined in terms of m1S and Λ. We run from mZ up to µ withthe SM fields, from µ up to Λ with the degrees of freedom of the magnetictheory and above Λ with the degrees of freedom of the electric theory.

A one-loop computation shows that the SU(3)c, SU(2)0,L and U(1)0,Y

couplings develop Landau poles at the scales

ΛL3 =m2S exp

( 2π

21α3(mZ)

)(mZ

µ

)− 13(µ

Λ

) 27( Λ

m2S

) 1621,

ΛL2 =m2S exp

( 2π

17α2(mZ)

)(mZ

µ

)− 19102(µ

Λ

) 2217( Λ

m2S

) 1117,

ΛL1 =m2S exp

( 2π

91α1(mZ)

)(mZ

µ

) 41546(µ

Λ

) 336546( Λ

m2S

) 215273.

(4.34)

We have taken λ1,2 ∼ 1 in the superpotential (4.17), so that m2S ∼ Λ/ε is thehighest scale in the electric theory, α1,2,3(mZ) are the U(1)Y×SU(2)L×SU(3)c


SM couplings evaluated at the Z boson mass mZ . In deriving eq.(4.34) wehave matched the SU(2)×U(1) couplings at the scale µ, using eq.(4.16) with

αm(µ) =2π

5log(

Λµ

) . (4.35)

Notice that the scale of the poles does not depend on m1S, since it cancels outin the contributions coming from Sij and Mij. Demanding for consistencythat ΛL

i > m2S constrains ε = µΛ

to be not too small. This is welcome from aphenomenological point of view, since a too small ε leads to a parametricallyweakly coupled magnetic sector (see eq.(4.35)) and too light magnetic vectorfields. On the other hand, ε cannot be too large for the stability of thevacuum, but values as high as 1/10 or so should be fine, given the estimateeq.(4.46): we address the issue of the lifetime of the vacuum in section 4.4.By taking natural choices for µ around the TeV scale, we see that all theLandau poles occur above m2S, with SU(3)c being the first coupling thatblows up, entering the non-perturbative regime in the 102 − 103 TeV range.

The Yukawa couplings λ1,2 and λL,R in the superpotential eq.(4.17) andeq.(4.22) might also develop Landau poles. A simple one-loop computation,in the limit in which the SM gauge couplings are switched off, shows thatthese poles appear at scales much higher than those defined in eq.(4.34). Ina large part of the parameter space the Yukawa’s actually flow to zero in theUV. This is even more so, when the SM gauge couplings are switched on,due to their growth in the UV.

4.4 Lifetime of the Metastable Vacuum

In presence of the meson mass terms eq.(4.20), in addition to the ISS vacuumeq.(4.6), other non-SUSY vacua can appear [103]. They can be dangerousif less energetic than the ISS vacuum, since the latter can decay throughtunneling too quickly to them. These vacua do not appear in our model, sincethe superpotential does not include meson terms of the form M2

ab. Other non-SUSY vacua, if present, can be found at qnm ∼ qni ∼ Mij ∼ Mnm ∼ Min ∼ µ,qn5 = 0, Mn5 = 0, Mi5 = 0, while M55 is still a flat direction. They do notlead to the desired pattern of symmetry breaking and they do not allow usto embed the SM in the flavor group. All these vacua have however exactlythe same tree-level energy of the ISS vacuum and would be irrelevant for thetunneling rate.


Supersymmetric vacua5 are expected when the mesons get a large VEV,in analogy with [101,103]. The scalar potential has a local maximum at theorigin in field space, with energy VMax = 5µ4, while at the local minimumVMin = µ4. We look for SUSY vacua in the region of large meson values,|Mij| � µ, |Mab| � µ. For simplicity, we take

〈Mab〉 = X δab , 〈Mij〉 = Y δij , Mia = 0 . (4.36)

For |X|, |Y | � µ, the magnetic quarks are all massive and can be integratedout. Below this scale, we get a pure SUSY SO(N) Yang-Mills theory witha set of neutral mesons M . The ISS superpotential admits N − 2 SUSYvacua, their existence is guaranteed by an explicit computation of the Wittenindex [106,107]. The same result for the index does not hold here because themeson mass term modifies the behavior of the superpotential at large fieldvalues, therefore we should recompute the index on our own. Neverthelesswe explicitly find, inspired by the original construction, SUSY vacua. Themagnetic superpotential, below the induced quark masses, is

W = 2Λ−52 (detM)

12 − µ2Maa −

1

2m1M

2ij −

1

2m2M

2ia , (4.37)

where we neglect the elementary sector, that gives rise to subleading correc-tions. The first term is non perturbatively generated by gaugino condensationand its dependence on detM is obtained by scale matching. By imposingthe vanishing of the F -term conditions, we find SUSY vacua at

X =Λ56µ−

13m

121 = ε−

13

√Λm1 = ε−

56√µm1 ,

Y =Λ512µ

56m− 1

41 = ε

56 Λ( Λ

m1

) 14

= ε−512µ( µ

m1

) 14,

(4.38)

whereε =

µ

Λ(4.39)

is a parametrically small number. The vacua eq.(4.38) can also be founddirectly in the electric theory. In the region where Sij is non-vanishing, allthe quarks Q are massive and the theory develops an Affleck-Dine-Seibergsuperpotential of the form [108]

Wnp = (N − 2−Nf )

(Λ3(N−2)−Nf

detM

) 1(N−2)−Nf

, (4.40)

5By supersymmetric vacua we mean those that are SUSY in the limit where we switchoff the external source of SUSY breaking.


where now M = MIJ = QIQJ . It is straightforward to check that this terminduces in fact the SUSY vacua eq.(4.38). The vacuum eq.(4.38) lies in therange of calculability of the magnetic theory if

µ� |X|, |Y | � Λ . (4.41)

The conditions eq.(4.41), together with the requirement that the mesons Mij

are not anomalously light, m1 ≥ µ, determine the allowed range for m1.Parametrizing

m1 = Λεκ , (4.42)

we get2

3< κ ≤ 1 . (4.43)

We now want to estimate the lifetime of the metastable vacuum, namelythe decay rate per unit time and per unit volume to the true vacuum givenby eq.(4.38). It is given by

Γ

V T∼ e−Sb (4.44)

where Sb is the bounce action, the euclidean action computed on the solutionof the e.o.m with proper boundary conditions. These boundary conditionsare such that there is one direction in field space interpolating among thefalse vacuum far in the past and exactly reaching at some finite time the“top of the hill”, the maximum of the potential between the two minima,the metastable one and the truly stable one. Heuristically we can think tothis transition happening in a localized region of space, whereas far fromthis “bubble” of true vacuum the field is still in the metastable vacuum.If the energy difference between the two vacua is small compared to theother energy scales the bubble is delimited by a “thin wall”, thin comparedto its radius [109]. In the present case the true vacuum has exactly zeroenergy, being SUSY, and therefore the difference is exactly the energy of themetastable minimum eq.(4.6): the thin wall approximation is not valid. Infact

VMax ∼ VMin ∼ VMax − VMin ∼ µ4 , VSUSY = 0 . (4.45)

As a very crude estimate of the lifetime of the metastable vacuum, wecan parametrize the potential using the triangular approximation [110], ne-glecting the direction in field space along the Y direction, which is alwayscloser to the ISS vacuum, given the bound (4.43).


The bounce action is parametrically given by [101,109,110]

Sb ∼|X|4

VMax

∼ ε−163

+2κ & ε−103 . (4.46)

We conclude that for small ε the metastable vacuum is parametrically long-lived and a mild hierarchy between µ and Λ should be enough to get a vacuumwith a lifetime longer than the age of the universe.

4.5 RG Flow of Soft Terms

In this section we explain, following [111, 112], how to understand the fateof UV soft terms in a SUSY gauge theory at strong coupling6: this is anessential ingredient for the model presented in chapter 5. For concretenesswe focus here on SO(N) gauge theories with N−2 < Nf ≤ 3/2(N−2) flavorsin the fundamental, admitting a Seiberg dual IR free description. This is thecase of interest for us, but what follows has clearly a wider applicability.More specifically, we want to determine the form of the IR soft terms inthe magnetic theory in terms of the electric ones. We first consider thecase with no superpotential: Wel = 0. Soft terms can be seen as the θ-dependent terms of spurion superfields whose lowest components are thewave-function renormalization of the Kahler potential and the (holomorphic)gauge coupling constant. The Lagrangian renormalized at the scale E is

Lel =

∫d4θ

Nf∑I=1

ZI(E)Q†IeVelQI +

(∫d2θS(E)Wα

elWel,α + h.c.), (4.47)

where

ZI(E) =Z0I (E)

(1− θ2BI(E)− θ2B†I(E)− θ2θ2(m2

I(E)− |BI(E)|2)),

S(E) =1

g2(E)− iΘ

8π2+ θ2 mλ(E)

g2(E)(4.48)

6An alternative derivation of the results of [111, 112] has been formulated [113]. Wefollow the original papers because we have found easier in this way to estimate the correc-tions coming from superpotential effects, although a reformulation in terms of the flow ofconserved currents and of the would-be conserved R-symmetry should be possible.


are the spurion superfields that encode the B-terms BI , non-holomorphicmass terms m2

I and the gaugino mass mλ. When there is no superpoten-tial, the BI terms are irrelevant and can be set to zero. The Lagrangian ineq.(4.47) is invariant under a U(1)Nf symmetry under which

QI → eAIQI , ZI → e−AI−A†IZI , S → S −

Nf∑I=1

tI8π2

AI , (4.49)

where AI are constant chiral superfields and tI are the Dynkin indices of therepresentations of the fields QI , tI = 1 for SO(N) fundamentals. In terms ofthese spurions, one can construct the following RG invariant quantities:

ΛS = Ee−8π2S(E)

b , ZI = ZI(E)e−∫R(E) γI (E)

β(R)dR . (4.50)

In eq.(4.50), b = 3(N − 2)−Nf is the coefficient of the one-loop β-functionβ(R), γI are the anomalous dimensions of the fields QI , and R(E) is definedas S(E) in eq.(4.48), but in terms of the physical, rather than holomorphic,gauge coupling constant. In terms of ΛS and ZI , one can further constructa U(1)Nf and RG invariant superfield:

I = Λ†S

( Nf∏I=1

Z2tIb

I

)ΛS . (4.51)

In the far IR, the dynamics of the system is best described by the magnetictheory, whose degrees of freedom are the mesons MIJ = QIQJ , the dualmagnetic quarks qI and the SO(Nf −N + 4) magnetic vector fields Vm. We

can use the RG invariants I and ZI and dimensional analysis to write thelowest dimensional operators in the low-energy Lagrangian:

Lmag =

∫d4θ(cMIJ

M †IJ ZIZJMIJ

I+ cqIq

†IeVmag Z−1

I (∏J

ZtJbJ )qI

)+

∫d2θ(Sm(E)Wα

mWm,α +qIMIJqJ

ΛS

)+ h.c. ,

(4.52)

where

Sm(E) =1

g2m(E)

− iΘm

8π2+ θ2 mm,λ(E)

g2m(E)

(4.53)

is the magnetic version of the spurion S defined in eq.(4.48). As shownin [111], these terms are the leading sources of soft terms provided that


mI � Λ, condition that will always be assumed. The last term in thesecond row in eq.(4.52) is the induced superpotential in the magnetic theory.Demanding the invariance of W fixes the U(1)Nf charges of the dual quarks qIto be QI(qJ) = 1/b− δIJ . These, in turn, fix the Z-dependence of the Kahlerpotential term of the magnetic quarks. The coefficients cMIJ

and cqI are realsuperfield spurions, the IR analogues of the wave function renormalizationconstants ZI(E). A relation between IR and UV soft terms is achieved bynoticing that in the far UV (IR) the electric (magnetic) theory is free. Thisimplies that for sufficiently high E, we can identify ZI with ZI , neglectingquantum corrections, and identify m2

I(E) ≡ m2I with the physical UV electric

soft terms. Similarly, in the far IR, we can neglect the θ2 and θ4 correctionsinduced by quantum corrections to cMIJ

and cqI . We can then compute the IRsoft terms by working out the θ2 and θ4 terms in the Lagrangian (4.52). Thephysical non-holomorphic soft masses for the mesons and magnetic quarksare

m2MIJ

= m2I + m2

J −2

b

Nf∑K=1

m2K , m2

qI= −m2

I +1

b

Nf∑K=1

m2K . (4.54)

As can be argued from eq.(4.54), positive definite UV soft terms alwaysflow in the IR to tachyonic soft terms for some mesons and/or magneticquarks [114]. Indeed, the following sum rule holds:

Nf∑I,J=1

m2MIJ

+ 2Nf

Nf∑I=1

m2qI

= 0 . (4.55)

In our derivation we have tacitly taken the dynamically generated scale in themagnetic theory to coincide with the electric one. This implies that the sameΛS defined in eq.(4.50) should be expressed in magnetic variables, namely

ΛS = Ee−8π2

bS(E) = Ee−

8π2

bmSm(E) , (4.56)

where bm = 3(Nf −N + 2)−Nf . Identifying the θ2 components of eq.(4.56),we get

limE→0

mm,λ(E)

bmg2m(E)

= limE→∞

mλ(E)

bg2(E). (4.57)

Notice that the θ2 term of ΛS introduces B-terms coming from both the D-and F -components of the magnetic Lagrangian Lmag that precisely cancel


each other. This is evident by noticing that the holomorphic rescaling

MIJ → ΛSMIJ (4.58)

removes ΛS from the leading order Lagrangian (4.52).Let us now apply these considerations to our specific set-up. We assume

that the electric soft terms do not break the Gf symmetry, so we effectivelyhave two U(1) symmetries, respectively rotating the quarks Qa and Qi, andtwo different soft terms, m2

1Q†aQa + m2

2Q†iQi. We therefore use eq.(4.54)

and we obtain the result quoted in eq.(4.30). Similarly in section 5.1, wherewe consider a model with Gf = SO(5) × SU(4) and fully composite righttop with b = 12, with this formalism we immediately obtain the soft termsreported in eq.(5.7).

Let us now see the effect of having Wel 6= 0. For concreteness, considerthe following two terms,

Wel = mQaQa +1

2λQiQjSij . (4.59)

We promote m and λ to chiral superfield spurions in the spirit of consideringan external unspecified SUSY breaking mechanism:

m→ m(1 + θ2Bm) , λ→ λ(1 + θ2Aλ) . (4.60)

We can still set BI = 0 in eq.(4.48), their effect being a redefinition ofthe Bm and Aλ terms in eq.(4.60). We can also reabsorb in Bm and Aλthe effect of the field redefinition (4.58) that would induce additional B-liketerms proportional to the gaugino soft terms. The above U(1)2 symmetry isunbroken provided m and λ transform as follows:

m→ e−2A1m, λ→ e−2A2λ , (4.61)

with Sij invariant. Two further U(1)2 and RG-invariants can be constructedstarting from m and λ:

Im = m†Z−21 m, Iλ = λ†Z−2

2 λ . (4.62)

The leading order Kahler potential for the mesons and the magnetic quarksis still of the form (4.52), but now cMIJ

and cqI are unknown functions of


Iλ/(16π2) and of Im/I.7 These corrections are sub-leading provided thatm� Λ and the effective coupling λ/Z2, at some UV scale E where the the-ory is perturbative, is smaller than 4π. Both conditions can be satisfied inour models. In first approximation we can then neglect the superpotentialcorrections to the RG flow of the soft terms. Of course, even when tak-ing Wel = 0, the relations (4.54) and (4.57) are only valid in the strict UVand IR limits and with vanishing mixing and SM gauge couplings. We havenot estimated the corrections coming from relaxing the above approxima-tions, assuming they are sub-leading in eqs.(4.54) and (4.57). It would beinteresting to perform a more careful analysis to check the validity of thisassumption.

There is an important consequence in having a non-vanishing Wel. In theIR, the first term in eq.(4.59) becomes linear in the mesons Maa and the Bm

term induces a tadpole for these fields. This is nothing else that the defor-mation discussed in the next section, 4.6. The tadpole changes the vacuumstructure of the model, as we will fully show in the next section. Extremiz-ing the whole scalar potential, soft terms included, we will get eq.(4.67) andeq.(4.69), namely

〈qmn 〉 =µ√hδmn , 〈M55〉 = X5 , 〈Mmn〉 = Xδmn . (4.63)

The presence of soft terms in the composite sector affects also the analysisof the vacuum decay pursued in section 4.4. We have checked the bound onthe soft terms in the composite sector (eq.(4.29), with the addition of theB-terms in the composite sector) above which L��SUSY can no longer be takenas a perturbation of the SUSY scalar potential in the region of large mesonVEV’s, eq.(4.36). In particular, we have verified under what conditions thevacuum displacements from the SUSY values δX/X and δY/Y , where X andY are defined in eq.(4.36), are much smaller than one. Comparable boundsarise from the soft terms m2, mm,λ and Bµ2 . We get

|m2| ∼ |mm,λ| ∼ |Bµ2| � ε43−κ

2 |µ| , (4.64)

where κ is defined in eq.(4.42). Given the bound (4.43) on the allowed valuesof κ, we see that the soft terms are constrained to be parametrically smaller

7These functions are not completely unrelated, since the combination of Kahler termsassociated to conserved global currents should precisely match in the UV and IR theories[113]. We have not studied this flow in detail, since we anyway neglect the effects of suchcorrections.


than µ. We have numerically explored also the region of soft terms larger thaneq.(4.64), resulting in shifts δX & X, δY & Y . Although it is not possible todraw a definite conclusion from this numerical analysis, we believe that thebound (4.64) is quite conservative, since the would-be SUSY vacuum energybecomes greater than the one of the ISS-like vacuum (4.32), when the softterms become comparable to (or larger than) µ. This is intuitively clear bynoticing that the dominant source of energy coming from L��SUSY is the softterm m2

2X2 (being |X| � |Y |) and this is positive definite. If this is the case,

the ISS-like vacuum would become absolutely stable, provided that othernon-SUSY vacua with lower energy do not appear elsewhere in field space.

4.6 Deformation

In this section we study soft deformations including, besides scalar and gaug-ino masses as in eq.(4.29), A and B terms for the couplings in the superpo-tential eq.(4.3). We have just discussed the inclusion of SUSY breaking termsin the theory, in the previous section 4.5: the techniques employed to followthe soft masses [111, 112] cannot be used in the case of A, B terms becausethe former can be computed only in absence of any superpotential and thusthey are expected to be valid up to perturbative corrections in couplings,while the latter are identically zero if the superpotential vanishes.

The most general soft terms may lead to unwanted tachyonic directionsand we restrict to safe cases where they only modify the spectrum: thishappens if they are not too large with respect to the holomorphic and softmasses. As we already argued the presence of B terms for the electric quarksinduces a term of the form

Lsoft ⊇ −µ2Bµ2TrM (4.65)

which introduces a new qualitative feature, even for arbitrary small Bµ2 . Infact the scalar potential now includes

V ⊇ |2hMnnqnm|

2 + |hqnmqnm − µ2|2 +(µ2Bµ2Maa+h.c.)+ m22|Mnn|2− m2

4|qnm|2 .

(4.66)The VEV of the magnetic quarks, eq.(4.32), becomes

〈qmn 〉 =µ√hδmn , where µ '

√µ2 +

m24

2h−

hµ4(Bµ2)2√

µ2 +m2

4

2h(4hµ2 + 2m2

4 + m22)

2

(4.67)


expanding for small Bµ2 : the true value for µ is a solution of the equationcubic in µ2:

[µ2 − (µ2 +m2

4

2h)](m2

2 + 4hµ2)2

+ 2hµ4B2µ2 = 0 . (4.68)

At the same time magnetic mesons also acquire a VEV

〈M55〉 = X5 , 〈Mmn〉 = Xδmn where

X5 = −µ2Bµ2

m22

X = − µ2Bµ2

4hµ2+m22

. (4.69)

Contrary to the previous case the mesons Mab have a nonzero VEV. Noticethat the symmetry breaking is still of the form eq.(4.7), SO(4)m × SO(5)→SO(4)D. The spectrum of the theory gets modified but the only qualitativedifference is about the Goldstone bosons: the four uneaten ones, identifiedwith the Higgs, are now contained in the massless field combination

cosα Re qn5 + sinα Re M5n , where sinα =2(X −X5)

f. (4.70)

Their kinetic term comes from |Dµqa|2 and |DµMab|2; at the non linear levelthey are described by a σ-model through a matrix U as in eq.(3.18) with anew decay constant given by

f =

√2

h

(µ2 + 2h(X −X5)2) . (4.71)

In the limit of vanishing B terms we have X = X5 = 0 and cosα = 1.For numerical analysis we work in the regime of small Bµ2 , in particularwe neglect its effects on the Higgs mass: this is consistent as long as thesuppression between Bµ2 and other masses, both holomorphic and soft, is atleast of the order of one-loop effects.

4.7 Quark Masses

The plain generalization of partial compositeness in our SUSY setup to allquarks and leptons does not work: it requires the presence of a large numberof (super)partners, leading to tremendously large flavor symmetry of the


composite sector and aggravating the problem of SM Landau poles. Thereforewe abandon it for all the fermions but the top.

To induce all the SM Yukawas we need a further explicit breaking of theSO(5) global symmetry, proportional to two matrices λABU and λABD whereA,B = 1, 2, 3 are family indices. The extension to leptons through anotherpair of matrices is straightforward. Deformations in the electric superpoten-tial8

Wel ⊇λABUΛL

(ξiaL,U

)A(ξibU)BQaQb +

λABDΛL

(ξiaL,D

)A(ξibD)BQaQb (4.72)

generate Yukawa terms if the dual mesons Mab ∼ QaQbΛ

get a vev, eq.(4.69).ξL,U and ξU are the spurionic embeddings of up type quarks in a fundamentalof SO(5), as in eq.(3.13). ξL,D and ξD are the spurions for down type quarksand can be defined in analogy to the up case but with a different X chargeassignment, X = −1/3. The most general low-energy Lagrangian will contain

L ⊇ qALεuBRH

c + qALλABu uBRH

c + qLAλABd dBRH + ...+ h.c. (4.73)

where9

λu,d = λ†U,DΛ

ΛL

sinα (4.74)

and the dots stand for higher dimensional operators: eq.(4.73) arises fromthe expansion in powers of f−1 of Mab = Uca〈Mcd〉Udb once we make explicitthe Higgs dependence through the matrix U defined in eq.(3.18). H is theHiggs doublet

H =(H(+), H(0)

)t=

1√2

(ih1 + h2,−ih3 + h4

)t. (4.75)

Without loss of generality we can go to the top basis in which ε ∼ εALε∗BR

f2is

different from zero only for A = B = 3: this is the term generated by themixings in eq.(4.24). The second and the third terms in eq.(4.73) are thenew operators responsible for the other quark masses. They have similaritieswith technicolor theories, where quark bilinears are coupled to an operator H

8The simplest way to induce these operators, as well as the the ones in eq.(4.19), isthrough the exchange of heavy chiral superfields, schematically W = λABξAΦξB+QΦQ+ΛL

2 Φ2 as already explained in the main text.9The dagger is only for notational convenience.


arising from a strongly interacting theory and responsible for EW breaking.The main differences are first that in our case the Higgs is protected by ashift symmetry and second that this coupling is not the dominant source forthe top mass.

To estimate the size of these masses we restrict for a moment on a singlegeneration:

L ∼ λD sinαΛ

ΛL

v bRbL + h.c. (4.76)

where v = 246 GeV is the Higgs VEV and Λ is the crossover scale dynami-cally generated. If ΛL ∼ 10Λ the correct value for the bottom mass can bereached with λD = O(1) and sinα ∼ 0.1. As discussed before a natural valuefor Λ is around 10 TeV while ΛL can be chosen to be the scale of Landaupoles for SU(3)c, in the region 102 - 103 TeV. Other quarks require smallercouplings: we do not explain neither the hierarchy among SM masses northe hierarchical structure of the CKM matrix, we assume them and we onlydistinguish between the top, partially composite, and other quarks, elemen-tary. This different origin of the masses results in the splitting between thetop and other quarks, making them naturally live at two different scales.

Finally we define Yd = λd and Yu = λu + ε, the down- and up-type quarkYukawa matrices; they can be diagonalized with

Yu → Y diagu = V †uYuUu , Yd → Y diag

d = V †d YdUd . (4.77)

If we perform the transformations Vd, Uu and Ud we go to the basis in whichYd is diagonal, while Yu → V †d VuY

diagu and we can define the CKM matrix

VCKM = V †uVd. Thus

λu → V †d λuUu = V †CKMYdiagu − V †d εUu . (4.78)

Since the matrix V †d εUu has arbitrary entries the second term signals a de-parture from minimal flavor violation, as it is depicted in [115]. In the nextsection we elaborate on it and on its consequences.

4.8 Flavor Analysis of Dimension Five Oper-

ators

A number of processes involving transitions in flavor space, ∆F = 0, 1, 2,results in flavor and CP observables and they very often receive sizeable


contributions from the presence of new physics, which in a broad class ofCHM are mainly induced by the mixing of quarks with their partners. Sincefor the lightest generations the mixings are weaker generically we have aprotection mechanism, known as RS-GIM [116] (see also [117]).

Despite the fact that the mixings are related to SM Yukawas, the re-sulting suppression might be not enough for a generic composite sector andadditional flavor symmetries are frequently postulated: a recent review isprovided in [118]. If light quarks are not partially composite, that is thereare no mixings with bound states, these contributions are absent: this is thecase for the interactions introduced in subsection 4.7.

On the other hand our model exhibits the same tensions of the MSSM,due to the presence of sparticles around the TeV scale: squark mass matricescannot be completely anarchic. Solutions to regulate the contributions toflavor processes are either to assume a certain level of degeneracy or alignmentamong squarks masses or to rely on some hierarchy between the first twogenerations and the third one, without threatening the naturalness, endingup with a scenario close to effective SUSY, depicted for instance in [119]where a discussion on flavor processes is also present. Correlations amongnew physics contributions in different processes could help in the future todistinguish among these possibilities [120]. We derive our results with alignedsquarks masses10 and we allow for small misalignment treated in the massinsertion approximation.

We do not perform a full analysis of all existent bounds; we instead con-centrate in what follows on the effects of the physics leading to the superpo-tential in eq.(4.72): at the scale ΛL other operators are plausibly generated.Under the spurionic flavor group U(3)q×U(3)u×U(3)d we assign the quantumnumbers:

qL ∼ (3, 1, 1), ucR ∼ (1, 3, 1), dcR ∼ (1, 1, 3), λu ∼ (3, 3, 1), λd ∼ (3, 1, 3) .(4.79)

Compatibly with these charges, with gauge invariance and with the holomor-phy of the superpotential we can write the following dimension five opera-

10Alignement is nicely realized in a variety of SUSY breaking mediation schemes: forinstance in gauge mediation A-terms vanishes at the mediation scale.


tors11,12

W ⊇ a1

ΛL

(ucRλUqL) (dcRλDqL) +a2

ΛL

(ucRλU t

AqL

)(dcRλDt

AqL

)= (4.80)

=Y ABCD

ΛL

[a1

(ucR,AqL,B

) (dcR,CqL,D

)+ a2

(ucR,At

AqL,B

)(dcR,Ct

AqL,D

)]where Y ABCD = λABU λCDD . Dimension five operators in the MSSM are dis-cussed in [121–123]: they result in, among other terms, contact interactionsbetween two quarks and two squarks. We assign the couplings λu,d to thevertices quark-squark-higgsino, neglecting deviations for tops, stops and leftsbottoms. With higgsino exchange we draw one-loop diagrams contribut-ing to four fermions interactions, experimentally constrained by ∆F = 2transitions in mesons. The resulting operator is

(dR,CkdL,Dl)(dL,EidR,Fj) · (4.81)

· λEAu λBFd

(4π)2mΛL

{δijδkl

[Y ABCD(a1 −

a2

6)− Y ADCB a2

2

]−

B ↔ Dj ↔ l

},

where m is a common soft mass for the squarks and the higgsino in the loop.For the general case of different masses the loop integral results in the follow-ing function of squarks and higgsino masses, mQ, mu and mh respectively:

I(mh, mQ, mu) = −mh

m2hm

2Q log

m2h

m2Q

+ m2Qm

2u log

m2Q

m2u

+ m2um

2h log m2

u

m2h

(m2h − m2

Q)(m2Q − m2

u)(m2u − m2

h)(4.82)

such that

I(m, m, m) =1

2m, (4.83)

altough we use eq.(4.81) and we do not need this exact expression. Foroperators of the form

c

Λ2F

(dRdL)(dLdR) (4.84)

the most stringent bounds come from kaons (the strongest is on the CPviolating part). The Wilson coefficient computed from eq.(4.81) identically

11At high energies the flavor spurions are λU,D and not λu,d, the difference being a factorΛL

Λ sinα ∼ 100.12tA are the SU(3)c generators such that tAijt

Akl = 1

2 (δilδjk − 13δijδkl).


vanishes, even in the non MFV limit of eq.(4.78). Non aligned squark massesat ε = 0 results, in the mass insertion approximation, in (the hadronic matrixelement with j ↔ l is less significant by a factor of 3 [124,125])

c

Λ2F

' 1

(4π)2mΛL

A2λ5

(ΛL

Λ sinα

)4

ydysy2t δ

⇒

{ΛF = 1 TeV

c ' 10−8δ(

1TeVm

)(100TeV

ΛL

)(4.85)

where for concreteness we fix a1 = a2 = 1; A and λ are the parameters ap-pearing in the Wolfenstein parametrization of the CKM matrix and δ mea-sures the relevant misalignement of squarks: it is the mixing of the first two

families left handed squarks normalized with a common mass m2, δ =(m2

Q)1,2

m2 .From [126] we easily read:

Re c < 6.9× 10−9 , Im c < 2.6× 10−11 if ΛF = 1 TeV . (4.86)

Bounds from box diagrams for different processes, with squarks and gluinosat 1 TeV, are stronger, they set for δ an upper bound around 10−2, see [127]and references therein13, resulting in c ' 10−10, below the bound eq.(4.86)(the bound on the CP violating effect computed here is not fully satisfied,it needs a1 ' a2 = O(10−1) or so). The numerical value of c is of the sameorder also with general ε.

Similarly the up-type quarks are involved in D mesons oscillations: inthis case the calculation is performed in the up-type mass basis, that is

λu = Y diagu − V †u εUu , λd = VCKMY

diagd . (4.87)

The relevant operator has the same form as in eq.(4.81) with the exchangeu ↔ d. In this case the coefficient is identically zero only if ε = 0, and for

ε = O(1) it is controlled by(

ΛLΛ sinα

)2y2bAλ

2; its value can be recast as

ΛF = 1 TeV, c ' 10−10

(1 TeV

m

)(100 TeV

ΛL

)(4.88)

13Notice that in box diagrams down squarks run into the loop while loops with verticesfrom eq.(4.80) are sensitive to up squark mass insertions, constrained by box diagrams forD − D oscillation.


with a1 = a2 = 1, below the experimental constraints, c < 10−8 [126]. Smallsquarks mass insertions do not change this numerical value14.

Hence we can infer that the inclusion of eq.(4.80) does not reintroduceviolations and does not hack the solution settled to avoid flavor problems.

4.9 Numerical Analysis

We show now the numerical results from an extensive scan in parameterspace in the determination of the mass spectrum of the model, particularlythe Higgs mass. We fix εR by requiring the correct top mass Mtop(1 TeV) '150 GeV and then scan randomly for the other parameters searching forpoints with ξ ' 0.1. For any such point we then extract the Higgs mass fromthe exact potential and compute the full spectrum.

We find that the Higgs mass is distributed in the range 70 GeV .MH .160 GeV, peaking between 100 − 140 GeV. The measured value MH '126 GeV is therefore a typical value for this model. For each point of thescan we obtain the FT computing numerically the logarithmic derivative ofthe logarithm of the Higgs mass with respect to all the parameters of themodel, and taking the maximum value [19]. The FT ranges between ∼ 10and ∼ 300, the typical value being around 50, with no evident correlationwith the value of the Higgs mass.

Let us now discuss some properties of the spectrum in the gauge sectorand in the matter sector.

Gauge Sector

The mass of the spin-1 resonances is given by mρ = gmf , up to corrections oforder O(gSM/gm) due to mixing with the elementary SU(2)L ×U(1)Y gaugebosons. Considerations of metastability and perturbativity fix gm(f) ' 2.5,which means mρ ' 1950 GeV for ξ = 0.1. Such values are still abovethe experimental limits from direct searches at the LHC [57, 130], [59] forlimits not from experimental collaborations, but are in tensions with indirectbounds from the S parameter.

The lightest uncolored scalar resonance has a mass bounded from aboveby the same value as the vector resonance. It is usually the complex neutral

14Bounds on down-type squark mass mixings are reported in [128,129].


singlet M55 with a mass ∼ 1 − 1.4 TeV. With less frequency it is the Im qn5or the lightest eigenstate of the symmetric part of qnm.

Among the spin-1/2 states, the lightest ones in our scan are usually thewinos (200−1200 GeV), the doublets hu,d arising from qn5 and M5n (around 1TeV), or the fields in the (1,3) of SU(2)L×SU(2)R coming from the magneticgauginos ρ and the fermions in the antisymmetric part of qnm (in particular,the ones ρ±R with Y = ±1, 600 − 1300 GeV, which do not mix with theelementary bino and the ones ρ3

R which do mix, 200− 1200 GeV).The goldstino, contained in the superfield M55, combines with the gold-

stino coming from the external SUSY breaking: a combination of the twowill be eaten by the gravitino and the orthogonal will stay in the spectrum asa massive particle. The exact value for their masses depends on the F termsand we can have different mixed situations in collider experiments, leading toa cascades of decays from neutralino to pseudogoldstino in turn decaying tothe true goldstino, resulting in multiphoton events (and missing transverseenergy) [131].

Matter Sector

As in some non-SUSY CHM, the lightest colored fermion resonance is theexotic doublet with Y = 7/6: the singlet with Y = 2/3 coming from amixture of the elementary tR and Mi5 is heavier, typically ∼ 1 TeV, whilethe mass of the lightest fermion ranges up to 900 GeV.

In the case of colored scalars, the lightest one belongs almost always tothe fields which mix with the elementary tL and tR, respectively doubletswith Y = 7/6 and singlets with Y = 2/3. Actually the whole bidoublet (adoublet with Y = 7/6 and a doublet with Y = 1/6) is almost degenerate inmass, the mass difference between the two doublets being . 100 GeV. Themass of the lightest scalar ranges from 600 GeV to 1 TeV. See fig. 4.1 for ascatter plot.

The gluino has a mass M3 which does not enter the Higgs effective poten-tial at one-loop, therefore it can be heavier than the current bounds withoutaffecting the FT of the model: contrary to what happens in the MSSM theEW scale is only logarithmically sensitive to stops masses [2]. It is alsoworth mentioning the existence of the chiral superfield Mij with supersym-metric mass m1, singlet under the electroweak group and in the symmetriccomponents of the (3 + 3) × (3 + 3) representation of SU(3)c. Since it doesnot enter the Higgs potential at one-loop, its holomorphic mass m1 is free in


650 700 750 800 850 900400

500

600

700

800

900

1000

mferm !GeV"

mscal!GeV

"mh !GeV"

100

120

140

Figure 4.1: Masses of the lightest colored fermions and scalar resonances.In shaded green we superimposed the exclusions discussed in section 4.10.Colors represents Higgs mass as indicated aside.

our setup and for consistency we take it m1 < Λ ∼ 10 TeV. In the followingwe will assume that it is heavy enough and neglect its phenomenology.

4.10 Detection Bounds

Given the features outlined in the previous section direct searches shouldconcentrate on colored states, in particular fermions and scalars with exoticelectric charge 5/3. First we notice that there is a consistent R parity chargesassignement. We have defined the gauge sector as the one which contributesto the one-loop Higgs potential via the SM electroweak gauge couplings, whilethe matter sector as the one which contributes through the mixings ε. Thisclassification reflects also the R parity assignment for the superfields: it isthe same as the one of the corresponding SM superfield with which the fieldmixes.

This implies, as usual, that scalar colored partners and EW fermion part-ners are pair produced and that the lightest among them is stable. Fermionic


W,B,G ρm qna Mab qL tR Mia qni Mij

RP + + + + − − − − +

Table 4.2: RP assignment to the lowest component of the superfields.

top partners share the same R parity as elementary fields, because they mixwith them; since they are a typical signature of CHM models [52] dedicatedsearches exist: a bound on their mass from events with a pair of same signleptons has been put at 800 GeV by CMS [13]. For scalar particles thereare not such searches but we can reinterpret the results for sbottoms pairproduced and decaying into tops and charginos: events with two b-jets andisolated same sign leptons are considered by CMS in [132] and a bound isset at 450 GeV, well below the values found in the numerical scan, althoughthe analysis is not based on the full dataset available but only on a portioncorresponding to an integrated luminosity of 10.5 fb−1.

In the model presented the scalar with Q = 5/3 is, up to EW effects,degenerate with a full bidoublet of SO(4), namely with other scalars with 1/3and 2/3 electric charge. EW effects are Higgs VEV insertions, affecting themasses of these particles and removing this degeneracy inducing splitting oforder 100 GeV. Thus we can convey as indirect bounds limits on the masses ofthe other components of the bidoublet: in particular the scalar with Q = 2/3would behave similarly to a stop with decoupled gluinos. Bounds on stopsdecaying into top and neutralino or bottom and chargino in events with oneisolated lepton are derived by CMS from the full 19.5 fb−1 dataset and stopsare excluded with a mass approximatively below 650 GeV [133].

Also CMS collaboration provides a stronger bound, of 750 GeV [134], onpair produced stops decaying into tops and neutralino using razor variables.

Finally we stress that the simultaneous presence of fermions and scalarsin the same mass range can strengthen the respective exclusion limits. Alsoin our setup multiple scalar stop partners appear and each of these can beproduced and decay at the LHC thus heightening the number of expectedevents and consequently the exclusion bounds: this effect can be simplyestimated as follow. We denote with σ(M) the pair production cross sectionof one scalar top partner with mass M and with Mexcl,n the excluded massin case of n identical scalars; if we assume a BR = 1 in top and chargino we


800 1000 1200 1400 1600 1800 200010!5

0.001

0.1

10

1000

MX !GeV"

"!fb" scalar 8 TeV

scalar 14 TeV

fermion 8 TeV

fermion 14 TeV

Figure 4.2: Pair production cross sections at LHC through QCD interactions.Dashed lines are the LO values, computed with MADGRAPH 5 [135], usingCTEQ6L PDFs and the model produced with the package FEYNRULES [136];solid lines are the NLO, using a common KNLO = 1.5.

estimate

nσ(Mexcl,n) = σ(Mexcl,1) ⇒ Mexcl,n = σ−1(σ(Mexcl,1)

n) (4.89)

assuming that the production cross section for n particles is just n timesthe case with a single scalar in the spectrum: we neglect decay chains andmutual interactions which deserve a dedicated study. We numerically have

Mexcl,n −Mexcl,1

Mexcl,1

' 0.1 for n = 2, 3. (4.90)

Therefore we claim the following: the exclusion limits for n particles inthe spectrum with n = 2, 3 are O(10%) stronger then the limits obtained fora single particle.

Turning to non colored states the lightest particles are fermions withquantum numbers of EW gauginos or higgsinos. As recently summarizedin [137] limits on charginos and neutralinos pair produced have been set byATLAS [138] and CMS [139]: with all sleptons and sneutrinos decoupledthey set limits at 350 GeV from events with three or more leptons in the


800 1000 1200 1400 1600 18000.1

1

10

100

MX !GeV"

!!fb" 20 fb"1

75 fb"1

300 fb"1

8 TeV14 TeV

800 1000 1200 1400 16000.01

0.1

1

10

100

MX !GeV"

!!fb"

20 fb"1

75 fb"1

300 fb"1

8 TeV14 TeV

Figure 4.3: Expected exclusion bounds on masses of the lightest fermionic(left panel) and scalar (right panel) top partners from 75 (dashed line) and300 (solid line) fb−1 at

√s = 14 TeV. The dotted lines correspond to present

bounds at 800 and 750 GeV discussed in the text.

final state. This analysis also allows CMS to put bounds on sbottoms andexcludes at 95% CL masses below 570 GeV.

Projections for exclusion limits for scalar and fermionic top partners fromLHC at a center of mass energy of 14 TeV can be obtained simply rescalingintegrated luminosities15. Fig. 4.3 clearly shows that higher luminosities,and higher center of mass energies, data will probe the relevant part of theparameter space. We expect they will be able to exclude exotic 5/3 chargefermions up to 1400 (1650) GeV and scalars up to 1300 (1550) GeV withdata corresponding to an integrated luminosity of 75 (300) fb−1, assumingBRs = 1. For fermions we also expect single production to become moreimportant than pair production at these energies.

We conclude this section noting that we can interpret already existingexperimental searches to exclude portions of the parameters space of themain model described in this chapter, introduced in section 4.2: we expectfuture experiments, LHC at 14 TeV will play a preponderant role, to furtherprobe it and constrain it to regions with higher level of FT.

15ATLAS released projections for future sensitivities in [140].

Chapter 5

A Fully Composite Top RightCase

As stressed in subsection 2.4.1 models of composite Higgs in which the righttop entirely belongs to the BSM strongly interacting sector are genericallydisfavoured: the Higgs boson is predicted to be too light, not compatible withthe measured value for its mass. Nevertheless we find instructive to presenta simple realization of this idea, first introduced in [1], within the SUSYframework depicted in section 4.1: we do it in the following. Also the problemof SM Landau pole is less severe here than in the previous case, due to a morerestricted number of partners. We allow for an additional elementary field,color and weak singlet with an exotic hypercharge: despite the further explicitSO(5) breaking induced by the pairing with the corresponding composite fieldno reasonable corner of the parameter space is found.

5.1 A Fully Composite tR

We consider a SUSY SO(9) gauge theory with Nf = 9 electric quarks in thefundamental of the gauge group and an additional singlet Sij in the (1,10)of the global SO(5)×SU(4). We add to the superpotential (4.1) the followingterm:

λQiQjSij . (5.1)

The terms (5.1) do not break any global symmetry. The total anomaly-freeglobal symmetry of the model is

Gf = SO(5)× SU(4) . (5.2)

90

CHAPTER 5. A FULLY COMPOSITE TOP RIGHT CASE 91

In full analogy with the previous case of section 4.2 the theory turns to strongcoupling at a scale Λ, below which we rely on a magnetic description providedby Seiberg duality. In the magnetic theory eq.(5.1) turns into a mass termλΛM ijSij. If we take λ ∼ O(1) around the scale Λ, the singlets Sij and M ij

can be integrated out. At leading order in the heavy mass, this boils downto remove the chiral fields Sij and M ij from the Lagrangian. We summarizein table 5.1 the gauge and flavor quantum numbers of the fields appearing inthe electric and magnetic theories.

The mass spectrum is the same as given in section 4.1, with the exceptionof the multiplet M ij that has been decoupled together with the singlet Sij.The multiplet Mi5 is massless. We embed SU(3)c × U(1)X into SU(4) andSU(2)0,L×U(1)0,Y into SO(5)×U(1)X . The U(1)X is identified as the diagonalSU(4) generator not contained in SU(3)c, properly normalized, such that4 → 32/3 ⊕ 1−2 under SU(3)c × U(1)X . We identify tR as the (conjugate)fermion component of Mα5, α = 6, 7, 8. We also get an unwanted extrafermion, coming from M95. Being an SU(2)L singlet, ψM95 corresponds to anexotic particle with hypercharge Y = X = 2. We can get rid of this particleby adding to the visible sector a conjugate chiral field ψc that mixes withM95, in the same way as Mia is going to mix with tL. The field ψc is actuallynecessary for the consistency of the model, so that all gauge anomalies cancel.Consider for instance the cubic anomaly of the hypercharge, where once againY = T 3

R + X and SU(2)R ⊂ SO(4) ⊂ SO(5): the contribution from the SMfermions without the right top is proportional to A(SM − tcR) = 3(2/3)3;from the composite sector we have A(qni ) = 4[3(2/3)3 +(−2)3] and A(Mia) =5[3(−2/3)3 + 23]. Therefore, to have Tr[Y 3] = 0 we need to introduce a fieldwith Y = −2. With this addition this and all the other anomalies vanish.

In the UV theory, the mixing terms are

λtξiaQiQa + λφφ

iaQiQa . (5.3)

Like in the previous section, we have written the mixing terms in a formalGf invariant way by means of the superfields ξ and φ. These are spurions,whose only dynamical components are the SM doublet qL and the singlet ψc.


SO(9)el SO(5) SU(4)QNi 9 1 4

QNa 9 5 1

Sij 1 1 10

(a)

SO(4)mag SO(5) SU(4)

qni 4 1 4qna 4 5 1Mia 1 5 4Mab 1 14⊕ 1 1

(b)

Table 5.1: Quantum numbers under Gf and the strong gauge group of thematter fields appearing in the composite sector of model II: (a) UV electricand (b) IR magnetic theories.

More explicitly, we have

ξαa =1√2

bL−ibLtLitL0

2/3

, ξ9a = 0 , φαa = 0 , φ9a =

0000ψc

−2

, (5.4)

where we have omitted the color index in qL and ψc. In the magnetic theorythe Yukawa’s (5.3) become

εtξiaMia + εφφ

iaMia . (5.5)

Thanks to the last term in eq.(5.5), the multiplets M95 and ψc combineand get a mass εφ/

√2. The assumption of an external source of SUSY

breaking affecting only the visible sector cannot work now, because tR isa fully composite particle, and would result in an unacceptable light stoptR. We then also add SUSY breaking terms in the composite sector, byassuming that they respect the global symmetry Gf . In order to have awell-defined UV theory, we introduce positive definite scalar soft terms inthe electric theory and analyze their RG flow towards the IR following [111].The construction discussed in section 4.5, to whom we refer for all the detailson how it is performed and the underlying approximations, here is crucial.Neglecting soft masses for the magnetic gauginos and B-terms, the non-SUSYIR Lagrangian reads

−L��SUSY = m2L|tL|2 + m2

ψ|ψ|2 + (εLBL(ξL)iaMia +1

2mg,αλαλα + h.c.)

+ m21|Mia|2 + m2

2|Mab|2 + m23|qi|2 − m2

4|qa|2 ,(5.6)


where

m21 =

1

12(4 + 2ω)m2, m2

2 =1

12(−8 + 14ω)m2,

m23 =

1

12(−8 + 5ω)m2, m2

4 =1

12(−4 + 7ω)m2

(5.7)

are the soft mass terms for the scalars in the IR theory, determined in terms ofthe two SO(5)×SU(4) invariant soft terms in the electric theory, m2

1elQ†aQa+

m22elQ

†iQi, with m2 ≡ m21el and

ω =m2

2el

m21el

. (5.8)

As can be seen from eq.(5.7), there is no choice of ω for which all the magneticsoft terms are positive definite. If we take ω > 8/5, the first three terms inthe second row of eq.(5.6) are positive, while the last one is tachyonic. Thesetachyons are harmless, since the SUSY scalar potential contains quartic terms(both in the F and D-term part of the scalar potential) that stabilize them.Negative definite quadratic terms for the qa are already present in the SUSYpotential, resulting in fact in the vacuum eq.(4.6). The only effect of theLagrangian (5.6), at the level of the vacuum, is to change the VEV as ineq.(4.32), namely

〈qmn 〉 =µ√hδmn , where µ =

√µ2 +

m24

2h. (5.9)

The above treatment of soft terms as a perturbation of an underlyingSUSY theory makes sense only for soft terms parametrically smaller than Λ.Notice that we cannot parametrically decouple the scalars in the compositesector, while keeping the fermions at the scale µ, by taking the soft termsm2 in the range µ � m � Λ. This is clear from eq.(5.9), since in this limitwe decouple the whole massive spectrum in the composite sector. In orderto keep the compositeness scale around the TeV scale and avoid too lightscalars, we take the soft term mass scale around µ. In addition to that, westill have, like in the model in section 4.2, an “indirect” contribution to thecomposite soft masses coming from the mixing with the elementary sector, asgiven by eqs.(4.26) and (4.27). A linear combination of fermions given by tLand the appropriate components of ψMim

remains massless and is identifiedwith the SM left-handed top. The “goldstino” ψM55 is still massless in these


approximations. See the considerations made in the last paragraph of section4.2, that apply also here, for the possible mechanisms giving a mass to thisparticle.

5.2 Vacuum Decay

The non-supersymmetric vacuum we have found can be metastable and su-persymmetric (in the sense explained in footnote 5) vacua might appear, dueto non-perturbative effects in the magnetic theory. Contrary to the modelin section 4.2, we have not found SUSY vacua in the regime of validity ofthe magnetic theory. The only SUSY vacua we found appear in the electrictheory. Assuming Sij 6= 0 with maximal rank, i.e. 4, all electric quarks aremassive and the resulting theory develops the non-perturbative superpoten-tial eq.(4.40) where now N = Nf = 9. Taking the ansatz eq.(4.36) for thegauge-invariant meson directions and Sij = S0δij, we get

FX = −5Λ−6X32Y 2 +m = 0 ,

FY = −4Λ−6X52Y + λS0 = 0 ,

FS = λY = 0.

(5.10)

The only solution to eq.(5.10) is the runaway vacuum

Y → 0, S ∝ Y −73 →∞, X ∝ Y −

43 →∞ . (5.11)

We have found no other SUSY vacua at finite distance in the moduli spaceand we then conclude that the metastable vacuum eq.(4.6) is sufficientlylong-lived, if not absolutely stable.

5.3 Landau Poles

Landau poles for the SM gauge couplings at relatively low energies are ex-pected also in this model. Within the same approximations made in sub-section 4.3, a one-loop computation shows that the SU(3)c, SU(2)0,L and


U(1)0,Y couplings develop Landau poles at the scales

ΛL3 = Λ exp

( π

2α3(mZ)

)(mZ

µ

)− 74(µ

Λ

) 14,

ΛL2 = Λ exp

( 2π

9α2(mZ)

)(mZ

µ

)− 1954(µ

Λ

)2

,

ΛL1 = Λ exp

( 6π

305α1(mZ)

)(mZ

µ

) 41610(µ

Λ

) 236305,

(5.12)

where we have matched the SU(2) × U(1) couplings at the scale µ, usingeq.(4.16) with

αm(µ) =2π

3log(

Λµ

) . (5.13)

The presence of less flavors and singlet fields here with respect to the previouscase ofN = Nf = 11 in section 4.2, allows for a significant improvement in theUV behavior of α3, that now blows up at extremely high energies. However,the different embedding of U(1)X in the global group gives rise to severalfields with hypercharge |2| that significantly contribute to the running of α1.As a result, the first coupling to explode is now α1. For a sensible choice ofparameters, e.g. µ around the TeV scale and ε ∼ 1/10, we see that ΛL

1 isabout two orders of magnitude higher than Λ, around 103 TeV.

5.4 Numerical Analysis

The next step is the computation of the effective action for the Higgs field,performed in the unitary gauge. As we mentioned many times the Higgs isa NG boson of a spontaneous breaking of a global symmetry: the brokensymmetry is not exact and it is explicitly broken by the SM EW gauge groupand by the couplings εt and εφ. In the matter contribution to the Higgspotential we further distinguish between colored and non colored exotic fields.We then perform a numerical scan in the parameter space.

The value of εt is fixed by the top mass; the mixing εφ of the non colored

field is in principle free. γgauge, γ(c)matter and γ

(nc)matter are equally important

and they cancel against each other: the size of these cancellations is a lowerbound on the FT. For what concern the coefficient of the quartic term wehave β ∼ βmatter � βgauge.


The Higgs turns out to be too light (∼ 100 GeV) unless a sizable source ofSO(5) breaking comes from the non colored sector, as in fact was expected.

Since the Higgs mass square is proportional to the sum β(c)matter + β

(nc)matter in

principle raising the non colored contribution controlled by εφ would be suf-ficient. At the same time large values for εφ are disfavored because generally

γ(nc)matter < 0 and it tends to align the Higgs in a EW preserving vacuum. Due

to this tension the model as it stands is excluded. We have chosen to reportthe results because, despite SUSY, the construction is quite minimal andwe expect it to be representative for more general examples: it embodies acomposite top right model with the addition of an extra massive singlet. Thesituation can be improved if we introduce more FT: due to the logarithmicdependence on soft masses we would need stops at a scale O(100) TeV, def-initely loosing the naturalness. We can also introduce more complication inthe model or focus on SO(5) representations different from the fundamental,but we have not continued along this path.

Chapter 6

Conclusions and Outlook

In this thesis we collected the work of the author on SUSY CHM. We providedfour dimensional examples of BSM sectors both weakly and strongly coupled.The Higgs is included as a pNGB living on the minimal coset studied inliterature allowing for a custodial protection, SO(5)/SO(4). We introducedcouplings to the top field through partial compositeness and we focused ontop partners in the minimal representation of the global symmetry group, thefundamental 5. The EW gauge fields may or may not be semi-composite, weinvestigated both cases.

Here we briefly summarize the main results:

• SUSY has offered tools (Seiberg duality) to deal with strongly cou-pled gauge theories. This allowed us to approach questions about UVcompleteness of the models: we considered explicit models of Higgs aspNGB, both as a truly elementary scalar and as a composite object. Wereviewed how to constrain the most general form of soft SUSY break-ing masses in presence of confining theories, with the help of globalsymmetries, treating them as spurion superfields: this allowed to keepa well defined UV completion. At the same time SUSY has soften thedependence of the Higgs potential on a low energy cutoff.

• We extensively analyzed, both analyticaly and numerically, the Higgspotential, expressed in terms of UV parameters. We neglected theimpact of SM fermions different from top on the radiative potential,claiming they do not play a role in the EWSB. We have computed theFT of the models and shown that there exist regions in parameters

97

CHAPTER 6. CONCLUSIONS AND OUTLOOK 98

space tuned at 1 - 2 % level, reproducing the correct top and Higgsmasses.

• We proposed a mechanism different to partial compositeness to sup-ply masses to other quarks, and to leptons, through irrelevant oper-ators and we have considered the most stringent bounds from flavorprocesses. The introduction of dimension five operators generates sub-leading corrections to flavor observables therefore flavor violations canbe addressed as in more conventional SUSY theories, as the MSSM: weexplicitly focused on the case of aligned squark masses.

• We have shown which are the most important experimental bounds,mostly from direct searches of partners, on our models. The lightestcolored particles in the BSM spectrum are generically colored partnersof the top, both scalars and fermionic, and uncolored fermions withthe quantum numbers of EWinos and higgsinos. We expect these pre-dictions to be independent of the particular realizations we presented.Interestingly enough some versions of SUSY CHM can be soon acces-sible at Run II of LHC.

For other types of observables, like indirect measurements or deviationsof (Higgs) couplings we refer to the many results and estimates present inthe literature. LHC will soon let us know more about the detailed structureof the EWSB in nature: experimental collaborations have already achievedsignificant results constraining, from the available data, the simplest realiza-tions of BSM physics, probing both the MSSM and its variations and modelsof composite Higgs, as well as extra dimensional and other exotic theories.We hope that future efforts will help in shedding light upon less minimalmodels, as the class proposed and discussed in this thesis.

Appendix A

Renormalization of the HiggsPotential

A.1 Renormalization of the Higgs Potential

Here we want to show explicitly the cancellations mentioned in the maintext regarding the scale dependence of the effective potential of the Higgs,as imposed by eq.(3.5). We first focus on the case of the model with vectorresonances, focusing on the superpotential

Wmag = hqaMabqb − µ2Maa . (A.1)

The superpotential (A.1) is invariant under global SO(5) transformations.At tree-level, with a canonical Kahler potential, this leads to an SO(5) in-variant (and hence Higgs independent) F-term scalar potential. At the ra-diative level, however, the Kahler potential is renormalized and the gaugingof SU(2)L × U(1)Y explicitly breaks the SO(5) global symmetry. This im-plies that although holomorphy protects the superpotential (A.1) from quan-tum corrections, the physical, rather than holomorphic, coupling h splitsinto several components with different RG evolutions, depending on theSU(2)L×U(1)Y quantum numbers of the fields qa and Mab. For this reason,in order to better understand the one-loop behaviour of the scalar potential,it is convenient to directly start by the generalization of (A.1), where wedistinguish all the couplings h with a different one-loop evolution. Recallingthat q contains one singlet and two SU(2)L doublets with Y = ±1/2 and Mab

contains three SU(2)L triplets with Y = −1, 0, 1, two SU(2)L doublets with

99

APPENDIX A. RENORMALIZATION OF THE HIGGS POTENTIAL100

Y = ±1/2 and two singlets, we have:

Wmag =5∑i=1

hi(qaMabqb)

(i) − µ2Maa (A.2)

where i runs over all the five distinct possible combinations:

(10 · 10 · 10), (10 · 2±1/2 · 2∓1/2), (2±1/2 · 3∓1 · 2±1/2),

(2∓1/2 · 30 · 2±1/2), (2∓1/2 · 1′0 · 2±1/2) . (A.3)

We write the effective Higgs potential as

V (sh) = V (0)(sh) + V (1)(sh) + . . . . (A.4)

The tree-level potential V (0) is given by

V (0) = m21|qn5 |2 +m2

2|qnm|2 +5∑i=1

|FM(i)ab |

2. (A.5)

Notice that the D-term scalar potential , as well as the F-term potentialgiven by the dual quarks q manifestly vanish in the vacuum eq.(4.32), whereh is now identified with h5. The first two terms in eq.(A.5) are soft SUSYbreaking terms. In the vacuum, modulo an irrelevant constant term, theygive rise to a tree-level Higgs mass term of the form m2

Hs2hµ

2/h5, where

m2H = m2

1 −m22 . (A.6)

The mass term (A.6) violates the SO(5) global symmetry of the compositesector, which is assumed to be exact in the limit of vanishing mixing termsεi and SM gauge couplings. We assume in the following that mH at the scalef assumes a value of the same order of magnitude of the one expected to begiven by radiative corrections, namely g2/(16π2) in some mass unit. In thisway, we are justified of neglecting its effect in the tree-level potential.

The RG-invariance of the scalar potential at one-loop level is governedby eq.(3.5), which reads

∂

∂ logQV (1) + βλI

∂

∂λIV (0) − γnΦn

∂

∂Φn

V (0) = 0, (A.7)


It is easy to check that in the vacuum the last term in eq.(A.7) vanisheswhen we recover the SO(5) invariant limit with hi = h. The relevant β-functions entering in the second term of eq.(A.7) are βhi , βm2

1and βm2

2.1 We

have

βm21− βm2

2=

(3g2M22 + g′2M2

1 )

8π2, βh1 = h1(2γqS + γMS ) , ,

βh2 = h2(γqD1/2+ γMD1/2

+ γqS) , βh3 = h3(2γqD1/2+ γMT1 ) ,

βh4 = h4(2γqD1/2+ γMT0 ) , βh5 = h5(2γqD1/2

+ γMS ) , (A.8)

where

γqS = γq0 , γMS = γM0

γqD1/2= γq0 −

1

16π2

(3

2g2 +

1

2g′2), γMD1/2

= γM0 −1

16π2

(3

2g2 +

1

2g′2),

γMT1 = γM0 −1

16π2(4g2 + 2g′2) , γMT0 = γM0 −

1

16π2(4g2) (A.9)

are the anomalous dimensions of an SU(2)L singlet with Y = 0, an SU(2)Ldoublet with |Y | = 1/2, an SU(2)L triplet with |Y | = 1 and an SU(2)L tripletwith Y = 0. The factors γq0 and γM0 are the SO(5) invariant contributions tothe field anomalous dimensions, given by the couplings hi themselves and bythe SO(4) magnetic gauge interactions (for q only), that do not contributeto the Higgs-dependent scalar potential. They are reported in eq.(A.18).

After some algebra, the first term in eq.(3.5) reads

∂

∂ logQV (1) = − 1

16π2

1

2STr[M4] =

=1

16π2f 2s2

h

(µ2(3g2 + g′2)− (3g2M2

2 + g′2M21 )).(A.10)

It is straightforward to check that eq.(A.7) is indeed verified, when we takethe couplings hi all equal at the scale f :

h1(f) = . . . = h5(f) = h(f). (A.11)

The case of the model in section 3.3, without gauge resonances, is treatedin full analogy. The relevant superpotential is given in eq.(3.12), namely

W0(Z, q, ψ) = hZ(qaqa − µ2) +mψqaψa . (A.12)

1In principle, we also have βµ2 , but it anyway cannot contribute to the one-loop Higgs-dependent potential, since Maa is an SU(2)L ×U(1)Y singlet.


The couplings whose running are affected by the SM gauging are mψ and h.The decomposition analogous to eq.(A.2) is a sum of two contributions,

W0 =∑i

[hiZ(qaqa)

(i) +mψi(qaψa)(i)]− hZµ2 (A.13)

where i runs over the projections

(10 · 10) , (2±1/2 · 2∓1/2) . (A.14)

Without reporting the full computation we just quote the results for thebeta functions:

βh2 − βh1 = −3(3g2 + g′2)

16π2h , βmψ2

− βmψ1= −3(3g2 + g′2)

16π2mψ (A.15)

where we have subtracted the SO(5) preserving part. The third tree-levelcoupling to be considered is the Higgs soft mass mH , as defined in eq.(A.6).With these beta functions one can show that eq.(A.7) is satisfied.

In the next section we report the one-loop beta functions, computed withstandard methods, of the couplings introduced for the models with partiallycomposite top quark, namely the model without vector resonances introducedin section 3.3 and the version enlarged to include vector resonances, in theparametrization given in section 3.4.

A.2 SO(5) preserving anomalous dimensions

and β functions

In appendix A.1 we just used some one-loop beta functions for fields ap-pearing in the scalar potential of the main theories discussed. In particularsince we were interested in the scale dependence of the Higgs potential wefocused on the RG violating the SO(5) symmetry induced by SM gauging.Here we report the terms in the anomalous dimensions and in the beta func-tions which are SO(5) preserving. We quoted them in several points in themain text in discussions related to the behavior of certain couplings and thepossible appearance of Landau poles: here we list them for reference.

The model with elementary tR, without vector resonances, has the fol-lowing anomalous dimensions for the fields appearing in the superpotential


of eq.(3.11) and eq.(3.12):

γNL =1

16π2

(λ2L −

8

3g2

3

), γNR =

1

16π2

(λ2R −

8

3g2

3

),

γS =1

16π2

(5λ2

L −8

3g2

3

), γSc =

1

16π2

(5λ2

R −8

3g2

3

),

γq =1

16π2

(3λ2

L + 3λ2R + 4h2

), γZ =

1

16π2

(10h2

).

(A.16)

The β functions for the couplings appearing in the superpotential eq.(3.11)and eq.(3.12) are therefore given by:

βαλL =αλL2π

(9αλL + 3αλR + 4αh −

16

3α3

),

βαλR =αλR2π

(9αλR + 3αλL + 4αh −

16

3α3

),

βαh =αλ2π

(6αλL + 6αλR + 18αh

),

βα3 =α2

3

2π

(− 3Nc +

1

2Nfund

),

(A.17)

where in the last line Nc = 3 is the number of colors and Nfund = (3(1 +1 + 2))SSM + (2 + 10)extra = 24 is the number of fundamental representationsof SU(3)c in the theory, divided into SM ones and the extra ones from thetop-partner sector.

For what concerns the model with elementary tR with also vector reso-nances, specified by the superpotential eq.(3.27) and eq.(3.28), the anomalousdimensions of the fields are given by:

γNL =1

16π2

(4λ2

L

), γNR =

1

16π2

(4λ2

R

),

γXL =1

16π2

(5λ2

L − 3g2ρ

), γXR =

1

16π2

(5λ2

R − 3g2ρ

),

γq =1

16π2

(3λ2

L + 3λ2R + 12h2 − 3g2

ρ

), γZ =

1

16π2

(8h2).

(A.18)

From this we compute the β functions for the couplings in the superpo-


tential eq.(3.27) and eq.(3.28):

βαλL =αλL2π

(12αλL + 3αλR + 12αh − 6αρ

),

βαλR =αλR2π

(3αλR + 12αλL + 12αh − 6αρ

),

βαh =αλ2π

(6αλL + 6αλR + 32αh − 6αρ

),

βρ =αρ2π

(5αρ

).

(A.19)

In the anomalous dimensions and in the β functions we have neglected thecontribution coming from the SM gauge couplings being surely subleading inthe energy range of validity of eq.(A.18) and eq.(A.19), that is for energiesroughly between 1 and 10 TeV.

Appendix B

Unitarization of WW Scattering

In theories where the Higgs is a pNGB the 2 → 2 scattering amplitudesbetween the longitudinal polarizations of the W and Z bosons, and the Higgsitself, for energies higher than the Higgs mass grow quadratically with theenergy, A(E) ∼ E2/f 2, violating perturbative unitarity at a scale Λ ∼ 4πf .At this scale, or before, new degrees of freedom (in the form of either strongdynamics effects or new perturbative fields) must become important in thescattering to restore unitarity, namely to cancel the leading order dependenceof the amplitude on the total energy. In the following we will see how thefield content present in each of the two models described in section 3.3 andin chapter 4 is exactly what is needed to restore perturbative unitarity ofWW scattering, as expected in genuine linear models.

By means of the equivalence theorem [141–144] the amplitude of thescattering involving longitudinal polarization of EW bosons are expressedin terms of the eaten Goldstone, up to corrections O(MV

E), therefore we are

led to study the UV properties of

A(hahb → hchd) . (B.1)

In the SO(5)→ SO(4) coset, all hahb scattering amplitudes can be parametrizedin terms of only two functions of the Mandelstam variables, A(s, t, u) andB(s, t, u) [56]:

A(hahb → hchd) = A(s, t, u)δabδcd + A(t, s, u)δacδbd + A(u, t, s)δadδbc +

+B(s, t, u)εabcd. (B.2)

In our models, however, in the limit of zero SM gauging the gauge sector hasa PLR symmetry which fixes B(s, t, u) = 0. The NGB contribution to the

105

APPENDIX B. UNITARIZATION OF WW SCATTERING 106

scattering is universal and given by

ANGB(s, t, u) =s

f 2. (B.3)

The possible contributions to NGB scattering can be obtained simply bygroup theory and the fact that bosonic states must be symmetric under theexchange of identical particles:

hahb scattering: 4⊗ 4 = (1; J = 0)⊕ (6; J = 1)⊕ (9; J = 0) , (B.4)

where J is the total spin.

B.1 Minimal Model SO(5)→ SO(4)

In the theory introduced in section 3.3 the only NGB present are the fourcomponents of the Higgs doublet, ha, and there are no gauge bosons otherthan the SM ones. The gauge sector of the model is a supersymmetriza-tion of the linear σ-model presented in ref. [76] and in the Appendix Gof ref. [56]. The only term in the Lagrangian relevant to WW scatter-ing is the kinetic term of the real part of q = (φ + iφ)/

√2, which takes

a VEV 〈φ〉 = (0, 0, 0, 0, f). Expliciting the NGB dependence as φ(x) =

U(ha(x))〈φ〉(

1 + η(x)f

), where η(x) is a real singlet scalar field with mass

Mη =√

2hf , we can write the Lagrangian as

Lkin =1

2(∂µη)2 − 1

2M2

ηη2 +

f 2

4Tr [dµd

µ]

(1 +

η

f

)2

, (B.5)

where we defined the CCWZ structures [17], as in the general case of eq.(2.6),daµT

a + EaµT

a = iU †DµU and ∇µ = ∂µ − iEµ. The full NGB scatteringamplitude in this theory can be written as

A(s, t, u) =s

f 2

(1− s

s−M2η

), (B.6)

where the second term comes from the tree level exchange of the singletfield η. The total amplitude evidently recovers perturbative unitarity for√s�Mη.


B.2 Model with Vector Resonances

We now move to discuss unitarization in the model arising at low energy asSeiberg dual, see chapter 4 and in particular the gauge sector described insection 4.1 for the details on the Lagrangian. Now ten Goldstone bosons arepresent: six πA in the adjoint representation of SO(4)D and four ha in thefundamental of SO(4)D. In the unitary gauge, where the Goldstone bosonsin the adjoint are eaten by the ρ gauge bosons, the study of their scattering isshifted to the study of the ρρ scattering. With the aim to connect our studywith previous bottom-up studies of the effect of resonances in WW scatteringin CHM and their phenomenology at the LHC [56], in the following we willconcentrate only on the study of the scattering amplitudes among the fourNGBs which form the Higgs doublet.

All contributions to NGB scattering, see eq.(B.4), come from the kineticterm of the fields in the multiplet which takes a VEV triggering the sponta-neous symmetry breaking, in our case the real components of qna :

L = |Dµqna |

2 =∣∣iU †Dµq

na

∣∣2 = |i∇µq + dµq − gµqρµ|2 , (B.7)

where we used eq.(4.33) to render explicit the NGB dependence. The fieldsqna transform under the unbroken group SO(4)D ∼ SU(2)L × SU(2)R in therepresentations

qna : 1⊕ 9⊕ 6⊕ 4 = (1,1)⊕ (3,3)⊕ ( (1,3)⊕ (3,1) )⊕ (2,2) . (B.8)

Its decomposition in terms of component fields is

qna (x) =

(f√2

+η(x)

2

)δna +∆ALBR(x)(2TALTBR)na+

qAρ (x)√

2(TA)na+i

qn5 (x)√2δa5,

(B.9)where the singlet η and the symmetric traceless ∆ are complex, while theantisymmetric qρ and the fundamental q5 are real fields. From eq.(B.4) wesee that the only states which can contribute to the scattering are the singlet(η), the gauge bosons (ρ) and the symmetric (∆). Since we are interestedonly in the tree-level contribution to the scattering amplitude, we can studythem separately.

Let us start with the singlet η = (η1 + iη2)/√

2. Setting ∆ and ρµ to zero


in eq.(B.7) one can arrive easily to the Lagrangian1

L ⊃ |∂µη|2 +1

2Tr [dµd

µ]∣∣∣µ+

η

2

∣∣∣2 = (B.10)

=1

2

((∂µη1)2 + (∂µη2)2

)+f 2

4Tr [dµd

µ]

(1 +

η1

f+η2

1 + η22

4f 2

).

In the parametrization of ref. [56] it is easy to recognize aη1 = 12, aη2 = 0

and bη1 = bη2 = 14. From this we obtain the contribution of the η to the hh

scattering amplitude:

Aη(s, t, u) = −1

4

s

f 2

s

s−M2η

, (B.11)

where Mη =√

2hf . Setting to zero the scalars ∆ and η we can obtain thecontribution from the vector ρµ. The Lagrangian can be written as

L ⊃ f 2

4Tr [dµd

µ] +f 2

2Tr[(gρρµ − Eµ)2

], (B.12)

recognizing that, in the notation of ref. [56], aρ = 1. The contribution to thescattering amplitude which grows with the energy is

Aρ(s, t, u) = −3

2

s

f 2. (B.13)

The scalar ∆ = (∆1 + i∆2)/√

2 is a complex field in the (3,3) of SO(4). ItsLagrangian can be written as

L =∑i=1,2

{1

2Tr[(∇µ∆i)

2]−M2

∆i

2Tr[∆2

i ] + a∆ifTr[∆dµd

µ] + . . .

}, (B.14)

where, in components, ∆i = ∆ALBRi (x)(2TALTBR)ba and where the dots rep-

resent terms not relevant for WW scattering. In our case a∆1 = 1, a∆2 = 0and M∆1 = M∆2 =

√2hf . The scattering amplitude is given by

A∆(s, t, u) =(a2

∆1+ a2

∆2)

4

(s

f 2

s

s−M2∆

− 2t

f 2

t

t−M2∆

− 2u

f 2

u

u−M2∆

).

(B.15)

1Here we also used that δnc (TATB)cdδnd = δAB , δnc (T aT b)cdδ

nd = δab/2 and

δnc (TAT b)cdδnd = 0, where TA and T a are, respectively, the unbroken and broken gen-

erators of SO(5)→ SO(4).


For energies larger than the masses of all these resonances we have

Atot(s, t, u) = ANGB(s, t, u) + Aη(s, t, u) + Aρ(s, t, u) + A∆(s, t, u)' const.(B.16)

We see that, as expected, the exchange of heavy resonances restores unitaritybefore the scattering amplitude becomes non perturbative.

Appendix C

SO(N)

In this appendix we report our conventions for so(N) algebra, the same as [1].

so(N) = Span{tab, a, b = 1, . . . , N, a < b} where tabij =i

2[δai δ

bj − δbi δaj ] .

(C.1)The matrices tab satisfy

Tr[tabtcd] =1

2(δacδbd − δacδbd) (C.2)

[tab, tcd] =i

2(δbcδaeδdf − δbdδaeδcf − δacδbeδdf + δadδbeδcf )tef .

With N = 5 it is customary to define different generators:

T aL,R = −1

2εabctbc ∓ ta4

T 1L = t32 + t41, T 2

L = t13 + t42, T 3L = t21 + t43,

T 1R = t32 + t14, T 2

R = t13 + t24, T 3R = t21 + t34,

T a =√

2 ta5, a = 1, 2, 3, 4 .

It is straightforward to check that

Tr[T iA, TjB] = δijδAB , Tr[T aT b] = δab (C.3)

[T iA, TjB] = iεijkT kAδAB

[T a, T b] = −itab = iεabc(T cL + T cR

)+ iδb4(T aL − T aR)− iδa4(T bL − T bR)

where A,B = L,R. In this basis, T 1,2,3L generate SU(2)L and T 1,2,3

R generateSU(2)R of the SO(4) ∼= SU(2)L × SU(2)R local isomorphism. The matrices

110

APPENDIX C. SO(N) 111

T 1,2,3,4 generate the coset SO(5)/SO(4). The last line makes explicit thatthis coset is symmetric: the commutator among broken generators does nothave components along broken generators.

The SO(6) generators are taken to be, for N = 6,

T 1 = t32 + t14, T 2 = t31 + t42, T 3 = t12 + t43, T 4 = t16 + t52, (C.4)

T 5 = t51 + t62, T 6 = t36 + t54, T 7 = t53 + t64, T 8 =1√3

(t12 + t34 + 2t65),

T 9 = t36 + t54, T 10 = t14 + t23, T 11 = t24 + t31, T 12 = t16 + t25,

T 13 = t36 + t45, T 14 = t46 + t53, T 15 =

√2

3(t12 + t34 + t56) .

In this basis, T 1,...,8 generate SU(3)c. The U(1)X generator is given by(4/√

6)T 15, so that the fields ξL and ξR in section 4.2 have U(1)X charges2/3 and −2/3, respectively.

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